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Sri Sarada International Journal of Multidisciplinary Research Publisher

Chief Editor

Yatiswari Vinayakapriya Directress, Sri Sarada Educational Institutions, Salem. Email : [email protected]

Dr.S.Niraimathi, M.A.,M.Phil.,Ph.D.,M.Sc. Principal, Sri Sarada College for Women (Autonomous), Salem. Email : [email protected]

Sub-Editors: Dr.R.Thilakam, M.Sc.,M.Phil.,Ph.D. Head of the Department of Chemistry, Sri Sarada College for Women (Autonomous), Salem-16. Email : [email protected]

Dr.N.Prabavathi, M.Sc.,M.Phil.,Ph.D . Head of the Department of Physics, Sri Sarada College for Women (Autonomous), Salem-16. Email : [email protected]

Dr.C.Immaculate Mary M.C.A.,M.Phil.,Ph.D. Head of the Department of Computer Science, Sri Sarada College for Women (Autonomous), Salem-16. Email : [email protected]

Dr.M.K.Uma M.Sc.,B.Ed.,M.Phil.,Ph.D Head of the Department of Mathematics, Sri Sarada College for Women (Autonomous), Salem-16. Email : [email protected]

Dr.R.Uma Rani M.C.A.,M.Phil.,Ph.D. Associate Professor of Computer Science, Sri Sarada College for Women (Autonomous), Salem-16. Email : [email protected]

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Sri Sarada International Journal of Multidisciplinary Research

REVIEWERS Dr.N.J.Suthan Kissinger M.Sc.,M.Phil.,Ph.D Assistant Professor in Physics Department of General Studies Jubail University College Royal Commission for Jubail P.O Box 10074, KSA email:[email protected]

Thomas Vougiouklis Emeritus Professor of Mathematics, School of Education, Democritus University of Thrace Alexandroupolis, Greece. email: [email protected]

Dr. S. Kamalakannan Associate Professor Division of Biotechnology College of Environmental and Bio resource Sciences Chonbuk National University – Iksan Campus 194-195 Madong, Iksan City Jeonbuk, South Korea email: [email protected]

Dr. Asai Asaithambi Professor, School of Computing, University of North Florida, U.S.A. email: [email protected]

Dr.K.R.Nandagobal Associate Professor Department of Computer Science Institute of Technology Ambo University Ambo P.O.No.19 Ethiopia email : [email protected]

Dr. Dinesh Kumar Saini Associate Professor, Faculty of Computing and Information Technology Sohar University P.O.Box.-44, PC-311, Sohar, Sultanate of Oman email: [email protected]

Dr.G. Ganesan Professor, Department of Mathematics, Adikavi Nannaya University, Rajamundry, Andra Pradesh – 533 296. India. email:[email protected]

Dr.L.Guruprasad Assistant Professor Department of Physics, M.Kumarasamy College of Engineering Karur, Tamilnadu India. [email protected]

Sri Sarada International Journal of Multidisciplinary Research

Dr. Mandhakini Mohandas Assistant Professor Centre for Nanoscience and Technology Anna University Chennai – 600025 Tamilnadu India. email: [email protected]

Dr. P. Anitha Assistant Professor Department of Chemistry Government College of Engineering Salem – 11 Tamilnadu India. email: [email protected]

Dr.G. Thangaraj Professor and Head, Department of Mathematics, Thiruvalluvar University, Vellore, Tamil Nadu, India. email: : [email protected]

Dr.K.Pushpanathan Assistant Professor, Department of Physics, Government Arts College, Karur , Tamil Nadu, India. email: [email protected]

Dr.S.Ravichandran Assistant Professor of Physics, Sathyabama University, Jeppiar Nagar, Chennai , Tamilnadu, India email: [email protected]

Dr.S.Muthukumaran, Assistant Professor, Department of Physics, Government Arts College, Melur, Madurai DT, Tamilnadu, India email: [email protected]

Sri Sarada International Journal of Multidisciplinary Research TABLE OF CONTENTS S. No

1

2

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Title of the Paper

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GROWTH, CHARACTERISATION AND THEORETICAL STUDY OF UREA ALUMINIUM POTASSIUM SULPHATECRYSTAL FOR NLO APPLICATIONS N.Sudha Dr.R..Mathammal DEVELOPMENT AND ASSESSMENT OF PACLITAXEL LOADED JMCC-P (HEMA) NANOSCAFFOLDS FOR DRUG DELIVERY Dr.Elakkiya Thangaraju Sheeja Rajiv ON GENERALIZATIONS OF SOFT FUZZY GΔ SEMI-CLOSED SETS V. Visalakshi

1

5

11

STRUCTURAL AND OPTICAL CHARACTERISTICS OF PURE AND CADMIUM DOPED COPPER OXIDE NANOPARTICLES R.Esaivani Dr.S.S.Gomathi P.Monisha Dr.G.K. Vanathi Nachiyar INVESTIGATION OF MOLECULAR STRUCTURE AND SPECTRAL ANALYSIS OF PROPYL SALICYLATE BASED ON DFT CALCULATIONS Dr.R.Mathammal Dr.R.Hema Malini Tamil Selvi.P Mekala.R, Sangeetha.M SYNTHESIS, CHARACTERIZATION AND COMPUTATIONAL INVESTIGATION OFNI(II) COMPLEXOF N,N'- BIS(2-HYDROXYBENZYLIDENE)-1,2-DIAMINOBENZENE Dr.S .Anbuselvi E.Elavarasi Dr. V. Jayamani

20

26

34

7

DIGITAL IMAGE PROCESSING TECHNIQUES IN FARMING: A SURVEY Dr.R. Uma Rani P.Amsini

42

8

ON CHARACTERIZATIONS OF FUZZY ROUGH BI-MAXIMAL SPACES D. Vidhya

46

9

A VIEW ON INTUITIONISTIC PAIRWISE REGULAR-LINDELOF T. Ramya

10

11



SPACES

A VIEW ON FUZZY Ρ-HYPERCONNECTEDNESS IN FUZZY MULTISET TOPOLOGICAL SPACES Dr.B. Amudhambigai M. Rowthri V. Madhuri SOME PROPERTIES OF INTUITIONISTIC C ∗-MONOID SPACES C.Bavithra Dr.M.K.Uma Dr.E.Roja

49

56

63

Growth, Characterisation and Theoretical Study of Urea Aluminium Potassium SulphateCrystal for NLO Applications N.Sudha, Dr.R.Mathammal, aDepartment

of Physics ,Sri Sarada College for Women (Autonomous),Salem, Tamilnadu, India [email protected]

Abstract: Single crystal of Urea Aluminium Potassium Sulphate (UALUM) has been grown by slow evaporation solution growth technique. Powder X-ray diffraction study has been carried out to determine the lattice parameter. FTIR confirm the presence of functional groups. The absorption spectrum of the titled crystal is in the lower cut off wavelength lies around 300nm. Quantum chemical calculation of energy and geometrical structure of UALUM were carried out by using Gaussian 09 program by HarteeFock method at 3-21(d,p) level basis set. The calculation show that UALUM has NLO behaviour with non zerovalues.Kurtz and Perry technique also prove its nonlinearity. Keywords:Slow evaporation, HF calculation, NLO, Hyperpolarizibility 1. Introduction Semi organic based materials have great promise in electronics due to the ease of their design and synthesis to suit the requirements of optoelectronic technologists. In order to be useful in modern technology, the material should possess large second order opticalnon-linearity, short transparency cut-off wavelength and good thermal stability [1, 2]. In this era of photonics and lasers, organic molecules are capable of influencing photonic signal efficiency in technologies such as optical communication, optical computing and dynamic image processing. The versatility of semi organic materials is due to their superior properties such as higher susceptibility, faster response and flexibility [3–

5].Among organic crystals, urea and its derivatives have been studied very widely and it is found that these exhibit NLO properties [6, 7].In the past three decades, urea has been the subject of extensive theoretical and experimental study, mainly because of its interesting physical and chemical properties [8].In this paper, we are reporting the growth of Urea Aluminium Potassium Sulphate by slow evaporation technique. The grown crystal is characterised by powder XRD, UV-Vis and FTIR .The SHG property of the material was assessed by Kurtz Perry Techniques. The Molecular properties were calculated by Quantum Chemical Calculations. 1.1 Experimental Single crystals of Urea Aluminium Potassium Sulphate (UALUM) were grown by slow evaporation-solution growth technique. All the materials used in the synthesis were of analytical grade (E Merck; 98%) and used without further purification. Urea Aluminium Potassium Sulphate were taken in the ratio 1:1:1:2 separately dissolved in methanol and the solutions were thoroughly mixed together using mechanical stirrer for about an hour. The clear solution obtained was filtered off using whatman No.40 grade filter paper to remove the suspended impurities and kept aside without any mechanical shake for crystal growth in a dust free environment at room temperature. Single crystals of UALUM were collected carefully from mother liquor after 15 days. The harvested crystals were recrystallized repeatedly to get superior quality crystals as shown in Fig 1.

Fig1As Grown crystal of UALUM

Sri Sarada International Journal of Multidisciplinary Research Vol : 1, Issue: 1, Jan 2018

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1.2 Computational Analysis Quantum chemical calculations were performed using HF method at 3-21(d,p) level basis set. The optimised geometrical parameters, electric dipole,polarzibility, andhyperpolarizibility were calculated. The optimized structure is shown in Fig 2

Fig 3 Powder XRD pattern of UALUM(a) pure (b) dopant 2.2 UV spectral analysis

Fig 2Optimized structure of UALUM

2. Results and Discussion 2.1 Powder XRD Powder X-ray diffraction analysis isperformed for pure Urea crystals and for the dopantwhich does not reveal the presence of other phases. According to chemical bonding theory of single crystal growth [9,10] the well-defined and sharp peaks signify the good crystalline nature of the compound is shown in Fig 3.The crystal structure of UALUM has a tetragonal cell with four molecules in the unit cell.The values are shown in Table 1.

UV-Visible absorption spectrum of UALUM is recorded and the spectrum is shown in Fig 4.The compound exhibits a strong absorption band at 293 nm which is attributed to ᴨ to ᴨ* transition. This result reveals that charge transferis occurring in UALUM [11]. It is also to be noted that the absence of absorption above 300 nm is desirable for materials for NLO applications. Therefore, UALUM can be used as a SHG material in visible range above 300 nm.

Table 1 a b c α β γ V

11.06 Å 11.06 Å 13.56Å 90.00° 90.00° 120° 1437

Fig 4 UV spectrum of UALUM 2.3 Vibrational analysis The functional groups were identified by Fourier transform infrared studies using BRUKER 66V FTIR spectrometer. The FTIR spectrums of UALUM were recorded between 4000cm-1to 400cm-1.


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The spectrum is show in Fig 5.The N-H stretching vibrations of urea is observed at 3375 cm-1.The stretching vibration of carbonyl group is appeared at 1645cm-1.The peak at 595cm-1show the presence of NH2 deformation vibration of urea. The peak at 2946 cm-1may be due the presence of Potassium ion. The peaks at 2473cm-1 and 2086 cm-1 may be due to Al ion additives. The peak at 1096 cm-1 shows the symmetric SO2 stretching. The peaks at 974 cm-1 and 920 cm-1show so asymmetric stretching of SO. The peak at 696 cm-1 show SO symmetric stretching

β =[(βxxx+βxyy+βxzz)2+(βyyy+βyzz+βyxx)2 +(βzzz+βzxx+βzyy)2]1/2

100.0 95 2086

90

ULAUM - 1

85

output. The second harmonic signal of 39 mV was obtained for UALUM crystal, while the standard potassium dihydrogen phosphate (KDP) crystal gives a SHG signal of 11 mV for the same input energy. It shows that the SHG effect of nonlinearity of UALUM is 3.6 times that of standard NLO material KDP. 2.5 Hyperpolarisability Quantum chemical calculations for UALUM was carried out by Gaussian 09 by HarteeFock method at 3-21 G basis and the optimised structure was obtained [13]. The components of the first hyperpolarisability were calculated using the following equation:

80 75

The calculated value of β first hyper-polarisability is1177.1132 (ie) 3.429 times that of urea. The polarisibility of the title compound is calculated using the equation :

70 65 60 55

974

50 %T 45 920

40

2473

35

1645

30 25

696 595

20 15 10

2946

3375

1096

5 0.0 4000.0

3600

3200

2800

2400

2000

1800 cm-1

1600

1400

1200

1000

800

600

450.0

Fig 5 FT-IR spectrum of UALUM

2.4 SHG measurement The SHG effective nonlinearity of UALUM powder was determined using Kurtz Perry powder technique. It enables to measure the SHG effective nonlinearity of new materials relative to standard potassium dihydrogen phosphate (KDP)[12]. A Q-switched Nd:YAG laser operating at 1064 nm and 8 ns pulse width with an input repetition rate of 10 Hz and energy 31 mJ/pulse was used for this study. The second harmonic signal generated in the crystalline sample was confirmed from the emission of green radiation of wavelength 532 nm from the crystalline powder. The SHG output was converted into electrical signal and was displayed on a digital storage oscilloscope. The optical signal incident on photo multiplier tube was converted in to voltage

αtot=1/3[αxx + αyy+αzz] The calculated value ofαtotpolarisability is89.22.The value of dipole moment is found to be 10.2656 Debye. The high values of dipole moment, polarisibility, first hyper-polarisability are important factors for NLO property. 4.9 Molecular orbital calculations Molecular orbital calculations of the title compound were carried out with Gaussian 09w program using the HF method at 3-21(d,p) basis set predict the HOMO-LUMO energy gap. The frontier orbital HOMO-LUMO determine the way in which the molecule interacts with other species. The charge transfer character of the HOMO-LUMO transition is also demonstrated by the topology of these orbital and it is shown in Fig 6 and 7.The calculated energies are HOMO = -0.38842 energy LUMO = -0.0029 energy HOMO-LUMO ΔE = -0.38546 energy


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[9] D. Xu, D. Xue, J. Cryst. Growth 286 (2006) 113. [10] P. Yang, D. Xu, D. Xue, Acta Mater. 55 (2007) 5757. [11] A. Chandramohan, R. Bharathikannan, V. Kandavelu, J. Chandrasekaran, M.A.Kandhaswamy, Spectrochim. Acta A 71 (2008) 755–759 [12] S.K. Kurtz,T.T.Perry ,J Appl.Phys,39(1968) 3798-3813 [13]M.Karan,V.Balachandran,M.Murugan,M.K.Murali,A.Nataraj Spectrochim. Acta A 71 (2013) 84–95

Fig 6 HOMO

Fig 7 LUMO

5.Conclusion The results reveal that the crystal UALUM grown by SEST has good crystallinity. The functional groups is confirmed by FTIR. The UVVis spectrum promotes that the material has good transparency. SHG technique show that UALUM is 3.6 times greater of standard NLO material KDP. Further the DFT calculations like HOMO-LUMO shows that the material is chemically reactive. Hyperpolarzibility calculations prove that UALUM is 3.429 times greater than that of Urea. All these results promote that UALUM is NLO active material can be used for industrial applications. References [1] G.R. Desiraju, J. Mol. Struct. 656 (2003) 5–15. [2] D. Xue, K. Betzler, H. Hesse, D. Lammers, Solid State Commun. 114 (2000) 21–25. [3] T. Umadevi, N. Lawrence, R. RameshBabu, K. Ramamurthi, G.Bhagavannarayana, Spectrochim. Acta A 71 (2009) 1667– 1672. [4] S. Gowri, T. Umadevi, D. Sajan, S.R. Bheeter, N. Lawrence, Spectrochim. Acta A89 (2012) 119–122. [5] K. Jagannathan, S. Kalainathan, T. Gnanasekaran, N. Vijayan, G.Bhagavannarayana, Cryst. Res. Technol. 42 (5) (2007) 483– 487. [6] V. Chithambaram, S.J. Das, R.A. Nambi, S. Krishnan, Solid State Sci. 14 (2012) 216–218. [7] K. Kato, IEEE J. Quantum Electron. 16 (1980) 810–811. [8] D. Xue, K. Kitamura, J. Wang, Opt. Mater. 23 (2003) 319– 322.


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DEVELOPMENT AND ASSESSMENT OF PACLITAXEL LOADED JMCC-p (HEMA) NANOSCAFFOLDS FOR DRUG DELIVERY aElakkiya

Thangaraju* and bSheeja Rajiv

aDepartment

of Chemistry, Sri Sarada College for Women (Autonomous), Salem - 16, Tamilnadu, India bDepartment of Chemistry, Anna University, Chennai - 25, Tamilnadu, India *[email protected]

Abstract The main objective of this study was the release of Paclitaxel from Jute Micro-crystalline cellulose (JMCC) reinforced Poly-(2Hydroxyethylenemethacrylate) (p(HEMA)) nanocomposite by solvent casting method. JMCC was prepared from chemically modified jute by acid hydrolysis. Different weight percentage (1, 1.5 and 2 wt %) of JMCC was reinforced into p(HEMA), a hydrogel by solvent casting method to prepare JMCC-p(HEMA) nanocomposite. Scanning electron microscopy images analyze the extracted cellulose range and in addition the roughness of the JMCCp(HEMA) nanocomposite surface. The XRD pattern indicated that the increases of crystallinity with the increase in the amount of JMCC and decreases with increases in the amount of Paclitaxel. The interaction between the JMCC and p(HEMA) in the nanocomposite was studied by FTIR. The TGA results confirmed the JMCC-p(HEMA) nanocomposite was less stable compared to the raw p(HEMA) because of the broken of hydrogen bonds and crystallite structure of polymers. The water uptake percentage was increased on the addition of JMCC to p(HEMA) nanocomposite was found to be approximately 82 %. The percentage cumulative release of Paclitaxel from the JMCC-p(HEMA) nanocomposite at the end of 8th day was found to be approximately 78 % respectively. Keywords: Jute, JMCC, p(HEMA), Nanocomposite, Paclitaxel, Drug delivery Introduction Tissue engineering is a multidisciplinary field of research that applicable for the principles

of chemistry, engineering and life sciences toward the improvement of biological scaffolds that restore, and maintain the tissue functions. The fabricated scaffold must have sufficient mechanical strength especially elasticity for its desired application. The necessity of an excellent nanoscaffold is that it must be biocompatible and suitable to immune reaction during drug delivery and tissue engineering. Natural and synthetic biopolymer nanocomposites are very important scaffolds in biomedical applications especially drug delivery. Nanoscaffolds prepared from hydrogels propose a great platform for drug delivery due to their hydrophilic nature and tissue like flexible properties [1]. Hydrogel polymers are hydrophilic in nature that is insoluble in water but its swell in aqueous solution. p(HEMA) is an effective polymer to used for biomedical engineering applications due to its physical properties can be easily manipulated through formulation chemistry [2]. p(HEMA) has been developed as a carrier of water soluble anticancer drugs due to its non-toxicity, well – tolerated safety, good biocompatibility and non – antigenic properties [3]. Generally, the filler reinforced composites have the advantage of higher mechanical strength and structural integrity and also the controlled effective properties can solve problems of different applications [4]. Jute is a natural biodegradable fiber containing three main categories of the chemical compounds namely cellulose (58-63 %), hemicellulose (20-24 %) and lignin (12-15 %) [5]. The higher crystallinity of the jute fiber (73.4 %) indicates its suitability in the preparation of microcrystalline cellulose. The jute nanofiber was reinforced to biopolyester composites improved the mechanical properties [6]. The light weight and high tensile strength of cellulose nanocrystals was also prepared in the simple method [7]. The addition of biodegradable jute fibres (< 2 wt %) will increase the mechanical properties of the nanocomposites. Many researchers have reported the jute fiber – matrix increases the tensile strength, and stiffness of the jute reinforced composites [8]. Also, it can be used as fillers due to its renewability, biodegradability, excellent reinforcing ability, and high surface area [6]. Vilaseca et al investigated the partial


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delignification of jute strands by means of sodium hydroxide treatment [9]. The alkali treatment results increase in the strength and stiffness of the jute strand / starch composites. The higher extension of the hydrogen bonds at fiber-matrix interface gave higher strength and stiffness of the jute strand. The incorporation of biodegradable jute into the p(HEMA) nanocomposite increases the mechanical strength and better thermal properties even at very low reinforcement content ( 5 wt %) of the fillers [10]. The main scope of the study was the use of JMCC reinforced p(HEMA) hydrogel nanocomposite as a drug delivery system for paclitaxel. Micro-crystalline cellulose was extracted from jute fiber on acid hydrolysis and alkaline treatment. JMCC (1, 1.5 and 2 wt %) was added into p(HEMA) solution and was fabricated as nanocomposite by solvent casting method and evaluated for application as anti-cancer scaffold in the clinical arena. The water uptake measurements further confirmed the suitability of the prepared nanocomposite for drug delivery applications. The loaded Paclitaxel in JMCCp(HEMA) nanocomposite enables the release of Paclitaxel in aqueous phase. The release rate of Paclitaxel was strongly controlled by the nanocomposite architecture and the enzymes were highly active after the release. Materials and Methods Materials Jute fiber was collected from nearby local market, Chennai, India. Poly-(2Hydroxyethylenemethacrylate) (p(HEMA)) (Mw =1, 00,000) and Paclitaxel, were purchased from Sigma – Aldrich, India. Hydrogen Peroxide (H2O2), Formic acid, Dimethyl sulfoxide (DMSO), Dimethyl Acetamide (DMAc), Lithium Chloride (LiCl), Sodium hydroxide (NaOH), and Sulfuric acid (H2SO4) were purchased from Sisco Research Laboratories Private Limited, India. All chemicals were used as received without further treatment or purifications. Experimental Procedure The jute fiber was cut into small pieces and refluxed into 200 ml of formic acid (90 % (v/v)) for 2 hours. After that the fiber was filtered and

washed with formic acid, followed by hot distilled water. The fiber was further delignified with peroxyformic acid (PFA) (90 % formic acid with 4 % H2O2) at 80 °C for 2 hours. Then the resultant fiber was filtered and washed with formic acid. The above jute fiber (10 % pulp concentration) was taken and the pH was adjusted to 11 by adding NaOH. H2O2 (4 %) was added and heated to 80 °C for 1 hour and filtered and dried in an oven at 40 °C. The fiber was hydrolysed in 75 % of H2SO4 at 50 °C for 2 hours with constant stirring. Then the above was cooled and allowed to settle down for 1 day. After 1 day, the fiber was diluted with excess of distilled water and filtered; a white residue obtained was washed repeatedly with distilled water and using ammonium hydroxide solution (5 %). The resultant solution was diluted with more distilled water until it becomes acid free (near neutral pH). The residue was then dried in a vacuum oven and ground into a fine powder, which is designated as JMCC [11]. A mixture of JMCC was dissolved in DMAc was heated to 100 °C for 30 min in a round bottomed flask equipped with a narrow path condenser. Licl was added to the mixture and heated to 120 °C for 10 min [12]. After 10 min, the reaction mixture was stirred overnight for complete dissolution at 50 °C. JMCC (1 wt %, 1.5 wt % and 2 wt %) was added to p(HEMA) (5wt %) and stirred continuously for 5 hours at 50 °C. At the completion of 5 hours, the viscous solution was casted into already cleaned and dried petri dish and it was dried in an oven for 6 hours at 80 °C. The clean transparent thin film obtained was dried. JMCC (2 wt %) – p(HEMA) solution with Paclitaxel (0.5, 0.75, 1 wt %) was added and follow the above procedure to obtained Paclitaxel loaded JMCC- p(HEMA) nanocomposites. Characterization of Nanocomposite Scanning electron microscopy FEI Quanta FEG 200 – HRSEM was used to analyze the surface morphology of the prepared JMCC and JMCC-p(HEMA) nanocomposite samples. X-ray diffraction The crystallinity of the prepared JMCC and JMCCp(HEMA) nanocomposite were examined using X’pert pro PANalytical Instrument using Cu Kα radiation (λ=1.5418 Å).


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FTIR Spectroscopy The functional groups of the p(HEMA) powder, extracted JMCC and JMCC-p(HEMA) nanocomposite were characterized by FTIR spectra on Perkin-Elmer spectrophotometer RX 100 in the range of 400 cm-1 to 4000 cm-1. Thermal Properties The thermogravimetric analysis (TGA) of the extracted JMCC and JMCC-p(HEMA) nanocomposite were analyzed by TGA/DTA model SDT 2600. Differential scanning calorimetry of the prepared JMCC and JMCC-p(HEMA) nanocomposite were performed with NETZSCHGeratebu model DSC 200PC. Water Uptake Test The water uptake characteristics of the prepared JMCC-p(HEMA) nanocomposites in deionsed water was measured. Approximately 5.2 mg of completely dried nanocomposite was weighed and then immersed into 15 ml deionised water. Water uptake study was performed at 37 °C in an incubator after attaining the equilibrium state. The swollen samples were withdrawn from deionised water and weighed after gentle surface wiping with (Whattman No 1) filter paper. The equilibrium water uptake value (WUV) was calculated according to the following equation as follows.

Where, Ws is the water uptake weight of the nanocomposite at equilibrium and Wd is the dry weight of the nanocomposite [13]. In-Vitro Drug Release Studies In-vitro drug release studies were carried out with three different weight percentage of paclitaxel loaded JMCC-p(HEMA) nanocomposites. For each of the formulations 50 mg of the drug loaded nanocomposite was placed in 5 ml phosphate buffer solution (PBS) (pH 7.4) containing 10 vol % dimethylsulfoxide (DMSO) solution with the temperature kept at 37 °C in an rotating incubator shaker at 100 rpm. At regular time intervals, one milliliter of aliquot was removed and replaced with the same quantity of the mixture of buffer and DMSO solution. The amount of drug in

releasing media was determined by high performance liquid chromatography (HPLC) equipped with an ultraviolet (UV) detector at 227 nm [14]. Results and Discussion Surface morphological Studies Figure 1 shows the SEM images of JMCC and JMCCp(HEMA) nanocomposite respectively. The SEM image of JMCC confirm that the JMCC has appreciable reinforcing property and also in the 2 µm in range. The rough and uniform nanocomposite was observed in JMCC-p(HEMA) nanocomposites and also the roughness increases on the addition of JMCC into p(HEMA). It also illustrates the improvement of the interaction between the fiber and matrix in p(HEMA) composite by the addition of JMCC.

Fig. 1 SEM images of JMCC and JMCC-p(HEMA) nanocomposites X-ray diffraction The crystallinity of JMCC and JMCC-p(HEMA) nanocomposite were determined by XRD. The XRD profiles were shown in Figure 2. The diffraction pattern of JMCC showed a high intensity sharp crystalline peak at 2=22 ° which was the characteristic of isolated cellulose from jute. JMCC has high tensile strength due to its higher crystallinity [11]. The diffraction peak at p(HEMA) was found to be 2θ=24 °. p(HEMA) was thoroughly mixing with JMCC and also easily intercalate into the interlayers by means of hydrogen bonding due to its hydrophilic nature [15]. Hence, the crystallinity of p(HEMA) was increased due to the addition of JMCC but the overall JMCC-p(HEMA) nanocomposite was in the semi-crystalline region which was suitable for drug delivery.


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Fig. 2 XRD graph for raw p(HEMA), JMCC and JMCC1-p(HEMA) nanocomposite FTIR Spectroscopy Figure 3 shows the FTIR spectra of p(HEMA) powder, JMCC and JMCC-p(HEMA) nanocomposite respectively. The JMCC-p(HEMA) nanocomposite exhibits the presence of carbonyl group of p(HEMA) at 1755 cm-1 and C-O stretching at 1249 cm-1 while the characteristic peaks of alcoholic – OH groups appeared at 3454 cm-1 (Stretching (-O) and 1076 cm-1 (bending –OH) [16]. The characteristic peak of JMCC can be attributed to the stretching of OH and C-H groups at 3368 cm-1 and 2901 cm-1. The band at 1644 cm-1 exhibits the vibration of absorbed water molecules and also to the carbonyl groups has a lower absorbance intensity ratio; this peak could be due to the presence of small amounts of hemicelluloses [17]. The absorption band at 714 cm-1 and weak shoulder at 760 cm-1 can be assigned to Iα (triclinic) and I𝛃 (monoclinic) cellulose [11]. The other characteristic peaks represent C-C and C-H vibrations of –CH3 and –CH2 groups respectively [18]. Hence, the above results clearly reveals that the JMCC was successfully reinforced into p(HEMA) nanocomposites.

Fig. 3 FTIR graph for JMCC and JMCC loaded p(HEMA) nanocomposites Thermal Properties The results of TGA and DTG graph of p(HEMA) powder, JMCC and JMCC-p(HEMA) nanocomposite were illustrated in Figure 4. JMCC weight loss at below 262 °C was due to loss of water which was the hydroxyl groups of cellulose. There were two major decomposition of JMCC was observed at 362 °C and 627 °C respectively while there were three major decomposition of JMCC-p(HEMA) nanocomposite was observed at 344 °C, 417 °C and 693 °C respectively. The initial degradation temperature of p(HEMA) showed that the degradation was due to random chain scission. However, the degradation temperature of JMCCp(HEMA) nanocomposites was increased slightly than JMCC due to more hydroxyl groups present in the nanocomposite. Hence, these results confirms the prepared JMCC-p(HEMA) nanocomposite was suitable for drug delivery.


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Fig. 4 TGA graph of JMCC and JMCC1-p(HEMA) nanocomposite The DSC graph of JMCC and JMCC-p (HEMA) nanocomposite were shown in Figure 5. DSC always gives the information about the glass transition temperature, melting temperature and degradation temperature but here it proves the obtained degradation temperature was suitable for drug delivery.

Fig. 5 DSC graph of raw p(HEMA), JMCC and JMCCp(HEMA) nanocomposite

Fig. 6 Water uptake test for JMCC-p(HEMA) nanocomposites in deioinsed water

Water uptake test and In-vitro drug release studies The percentage water uptake of JMCC-p(HEMA) nanocomposite in deionised water was shown in Figure 6. It was noted that the percentage water uptake was increased on the addition of JMCC. This percentage water uptake was due to the hydrophilic nature of the JMCC. The rate of release of Paclitaxel from JMCC-p(HEMA) nanocomposite was dependent on two main factors, namely, the thickness of the Paclitaxel loaded JMCC-p(HEMA) nanocomposite and the rate at which JMCCp(HEMA) degrades and allows for the optimization of Paclitaxel release within the body. In general the drugs can be released in a controlled manner with the first order kinetics. In the present study, the feasibility and potential of JMCC-p(HEMA) nanoscaffolds as a drug delivery vehicle for Paclitaxel release was investigated. The in-vitro release of Paclitaxel into PBS containing DMSO over a period of 8 days is shown in Figure 7. The Paclitaxel loaded JMCC-p(HEMA) nanoscaffolds showed controlled and sustained release of Paclitaxel (i.e. 78 % release) in 8 days. At the initial stage, there was small burst effect which could be due to the release of some loosely bound Paclitaxel on the surface. This loosely bound Paclitaxel might be released by a mechanism of diffusion through the aqueous pores on the surface created by the water uptake by nanoscaffolds immediately after being exposed. At the later stage, the Paclitaxel release was slow, whose rate was determined by the diffusion of PBS into the Paclitaxel loaded JMCC-p(HEMA) matrix, because the release of Paclitaxel from Paclitaxel loaded JMCC-p(HEMA) nanoscaffolds depended upon the diffusion path filled up by PBS. The percentage cumulative Paclitaxel release at the end of 8th day period for 0.5 and 1.0 wt % formulations was 78 % respectively. No significant difference in release was observed between nanoscaffolds containing 0.5 wt % and 1.0 wt % Paclitaxel in the conducted release period. Nanoscaffolds formulated drugs and their in-vitro release profile helped to understand the behavior of these systems in terms of drug release, and therefore its efficacy.


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4.

5. 6.

7. Fig. 7 Drug release studies for Paclitaxel loaded JMCC-p(HEMA) nanocomposites 0.5 %, 0.75%, and 1% Conclusion The present study focuses on the JMCC reinforced p(HEMA) nanocomposite for drug delivery. The rod like shape with microcrystalline range of cellulose was extracted by solvent casting method and confirmed by SEM. The functional group of JMCC and JMCC-p(HEMA) nanocomposite was studied by FTIR. The amorphous nature and thermal stability of the prepared JMCC-p(HEMA) nanocomposite were confirmed by XRD and TGA. The water uptake test confirms the hydrophilic nature of the nanocomposite which has great advantage for drug delivery. In-vitro drug release confirms the controlled release of paclitaxel from the JMCC-p(HEMA) nanocomposite. Hence, the overall studies prove the ease of preparation of the nanocomposite via solvent casting method and the possibility of the prepared JMCC-p(HEMA) nanocomposite to act as an drug delivery vehicle. Acknowledgement The authors acknowledge the Department of Biotechnology (DBT) for financial assistance (Project No. BT/PR15000/GBD/27/280/2010 Dated 01.02.2011). References 1. Jeffrey A Hubbell (1996), Journal of controlled release, 39, 305-313. 2. Blumstein Alexandre (1965), Journal of Polymer Science Part A, 3, 2665-2672. 3. Ging-Ho-Hsiue, Jan-An Guu, and Chinchen cheng (2001), Biomaterials, 22, 1763-1769.

8. 9.

10.

11. 12. 13. 14. 15. 16.

17.

18.

Kunal Das, Dipa Ray, Indranil Banerjee, N R Bandyopadhyay, Suparna Sengupta, Amar K. Mohanty, Manjusri Misra (2010), Journal of Applied Polymer Science, 118, 143-151. Mohanty A K, Mishra M, Hinrichsen G (2000), Macromolecular Materials and Engineering, 276, 1-24. Kunal Das, Dipa Ray, Chitrita Banerjee, N R Banyopadhyay, Saswata Sahoo, Amar K Mohanty and Manjusri Misra (2010), Industrial and Engineering Chemistry Research, 49, 2775-2782. Wadood Y Hamad (2017), Cellulose nanocrystals: Properties, Production and Applications, Wiley Publication, ISBN: 978-1-119-96816-0, 312. Zimmermann T, Ohler E P and Schwaller P (2005), Advanced Engineering Materials, 7, 1156-1161. Kunal Das, Dipa Ray, N. R. Bandyopadhyay and Suparna Sengupta (2010), Journal of Polymer and the Environment, 18, 355-363. Vilaseca F, Mendez J A, Pelach A, Llop M, Canigueral N, Girones J, Turon X and Mutje P (2007), Process Biochemistry, 42, 329-334. Sarwar Jahan M, Abrar Saeed, Zhibin He and Yonghao Ni (2011), Cellulose, 18, 451459. Lirong Yao, Changhwan Lee and Jooyong Kim (2011), Fibers and Polymers, 12, 197206. Mehrdad Kokabi, Mohammad Sirousazar and Zuhair Muhammad Hassan (2007), European Polymer Journal, 43, 773-781. Jingwei Xie and Chi-Hwa Wang (2006), Pharmaceutical Research, 23, 1817-1826. Rahul Singhal and Monika Datta (2000), Polymer Composites, 13, 887-890. Simonida Ljubisa Tomic, Suzana I Dimitrijevic, Aleksandar D Marinkovic, Stevo Najman and Jovanka M Filipovic (2009), Polymer Bulletin, 63, 837-851. Abeer M Adel, Zeinab H Abd El – Wahab, Atef A Ibrahim and Mona T Al-Shemy (2011), Carbohydrate Polymers, 83, 676687. Silverstein R M, Bassler G C and Morrett J C (1991), Spectroscopic identification of organic compounds, 5th edition, Wiley, New York.


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ON GENERALIZATIONS OF SOFT FUZZY Gδ SEMI-CLOSED SETS V. Visalakshi,

Department of Mathematics SRM University, Kattankulathur, Chennai. Tamilnadu, India. [email protected]

Abstract: In this paper, soft fuzzy Gδ semi closed set is introduced. Using soft fuzzy Gδ semi closed set, generalized soft fuzzy Gδ semi-continuous function, generalized soft fuzzy Gδ semiconnected space, generalized soft fuzzy Fσ semicompact space are introduced and studied. In this connection, properties and characterizations are established. Also soft fuzzy T-Fσ space is introduced and studied. Examples and counter examples are provided wherever necessary. Keywords : gsfGδs-closed; gsfGδs-continuous; gsfGδs-connected space; gsfGδs-extremally disconnected space; gsfFσs-compact space. AMS MSC2010 : 54A40, 03E72.

continuous and perfectly gfs-continuous functions. He also discussed gfs-connected space, gfsextremally disconnected space and gfs- compact space. By using the concept of generalized fuzzy semi closed sets [5] and, the concept of generalized soft fuzzy Gδ semi closed set is introduced. The concepts of generalized soft fuzzy Gδ semicontinuous functions, gsfGδs-irresolute functions, strongly gsfGδs-continuous functions, perfectly gsfGδs-continuous functions are introduced and interrelations among these functions are discussed with suitable examples. The concepts of gsfGδs-connected space, gsfGδs-extremally disconnected space, gsfFσs-compact space and soft fuzzy T-Fσ space are established. 1.2 : PRELIMINARIES Definition: 1.2.1[9] Let (X, T) be any soft fuzzy topological space and (λ,M) be a soft fuzzy set in (X, T). (λ,M) 

is said to be soft fuzzy Gδ set if (λ,M) =

 (λi,Mi)

i 1

where each λiT, iI, MiX. (λ,M) is said to be soft 

fuzzy Fσ set if (λ,M) =  ((1,X)- (λi,Mi))where each i 1

λiT, iI. 1.3 : SOFT FUZZY SEMI CLOSED SETS Definition: 1.3.1

1.1 : INTRODUCTION Zadeh introduced the fundamental concepts of fuzzy sets in his classical paper[11]. Fuzzy sets have applications in many fields such as information[6] and control [7]. In mathematics, topology provided the most natural framework for the concepts of fuzzy sets to flourish. Chang [3] introduced and developed the concept of fuzzy topological spaces. The concept of soft fuzzy topological space is introduced by [8]. Various properties of soft fuzzy topological space was discussed by [9]. [9] introduced the concept of soft fuzzy Gδset. Jin Han Park and Jin Keun Park [5] introduced and studied the concepts of generalized fuzzy semi-continuous, gfs-irresolute, strongly gfs-

Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. A mapping f : (X, T) → (Y, S) is said to be soft fuzzy continuous if the inverse image of every soft fuzzy closed set in (Y, S) is soft fuzzy closed in (X, T). Definition: 1.3.2 A soft fuzzy topological space X is said to be soft fuzzy connected if it has no proper soft fuzzy closed and soft fuzzy open set. Definition: 1.3.3 Asoft fuzzy set (λ,M) in a soft fuzzy topological space (X, T) is said be soft fuzzy semiclosed ( in short, sfs-closed ) if (λ,M) ≥ int ( cl ( λ,M) )). Afuzzy set (λ,M) in a soft fuzzy topological


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space (X, T) is said be soft fuzzy semi-open ( in short, sfs-open ) if (λ,M) ≤ cl ( int ( λ,M )). Definition : 1.3.4 Let (λ,M) be any soft fuzzy set in a soft fuzzy topological space (X, T). Soft fuzzy semi closure of λ is defined as sfscl ( ,M ) =   {(,N) / (,N)(,M) and (,N) is sfs-closed}. Definition: 1.3.5 Let (λ,M) be any soft fuzzy set in a soft fuzzy topological space (X, T). Soft fuzzy semi interior of (λ,M) is defined as sfsint ( ,M ) =  {(,N) / (,N)(,M) and (,N) is sfs-open}. Definition : 1.3.6 A soft fuzzy set (,M) in a soft fuzzy topological space (X, T) is called generalized soft fuzzy semi closed (in short, gsfs-closed) if sfscl ( ,M ) ≤ (,N) whenever (,M) ≤ (,N) and (,N) is soft fuzzy open. A soft fuzzy set (,M) is called generalized soft fuzzy semi open (in short, gsfsopen) if its complement (1,X)(,M) is gsfs-closed. Definition : 1.3.7 Let (X, T) and (Y, S) be any two fuzzy topological spaces. A function f : (X, T)  (Y, S) is called generalized soft fuzzy semi-continuous (in short, gsfs-continuous) if the inverse image of every soft fuzzy closed set in (Y, S) is gsfs-closed in (X, T).

Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. A function f : (X, T)  (Y, S) is said to be perfectly gsfs-continuous if the inverse image of every gsfs-open set in (Y, S) is both soft fuzzy open and soft fuzzy closed in (X, T). Definition : 1.3.10 A soft fuzzy topological space (X, T) is said to be generalized soft fuzzy semi connected (in short, gsfs-connected) if the only soft fuzzy sets which are both gsfs-closed and gsfs-open are (0,ϕ) and (1,X). Definition: 1.3.11 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. A mapping f : (X, T) → (Y, S) is 1

called soft fuzzy Gδ-irresolute if f ( µ,N ) is soft fuzzy Gδ set in (X, T) for each soft fuzzy Gδ set (µ,N) in (Y, S). 1.4 : GENERALIZED CLOSED SETS Definition : 1.4.1

SOFT

FUZZY

GSEMI

A soft fuzzy set (,M) in a soft fuzzy topological space (X, T) is called generalized soft fuzzy G semi closed (in short, gsfGs-closed) if sfscl (  ) ≤  whenever  ≤  and  is soft fuzzy G. A soft fuzzy set  is called generalized soft fuzzy F semi open (in short, gsfFs-open) if its complement (1,X)  (,M) is gsfGs-closed. Example : 1.4.1 Let X = [0, 1] and T = {(0,ϕ), (1,X), (λn,M)} n( x ) =

5n 2 (n = 2, 3, ....), x 7( n 2  1)

Definition : 1.3.8

where

Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. A function f : (X, T)  (Y, S) is said to be strongly gsfs-continuous if the inverse image of every gsfs-open set in (Y, S) is soft fuzzy open in (X, T).

 X, M={0,1}. Then (2,M) is gsfGs-closed in (X, T).

Definition : 1.3.9

Definition : 1.4.2 Let (X, T) be any soft fuzzy topological space. Let (λ,M) be any soft fuzzy set, generalized soft fuzzy G semi closure of λ is defined as


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gsfGs-cl ( ,M) =   {(,N) / (,N)(,M) and (,N) is gsfGs-closed}. Definition : 1.4.3 Let (X, T) be any soft fuzzy topological space. Let (λ,M) be any soft fuzzy set, generalized soft fuzzy F semi interior of (λ ,M)is defined as gsfFs-int ( ,M ) =  {(,N) / (,N)(,N) and (,N) is gsfFs-open}. Property : 1.4.1 Every gsfGs-closed set is gsfs-closed. Remark : 1.4.1 Every gsfs-closed set need not be gsfGsclosed as shown in the following example. Example : 1.4.2 Let X = [0,1] and T = {(0,ϕ), (1,X), (λn,M)} where n( x ) =

2n  3 (n = 1, 2, 3,......),  x X, M={0,1}. 5n  1

Then (1,M) is gsfs-closed but notgsfGs-closed in (X,T). Remark : 1.4.2 The intersection of any two gsfGs-closed sets need not be gsfGs-closed as seen in the following example. Example : 1.4.3 Let X = {a, b, c} and T = {(0,ϕ), (1,X), (λ,M)}where ( a ) = 1, ( b ) = 0, ( c ) = 0, M={a}. Define fuzzy sets (1,M1) and (2,M2) in (X, T) as follows: 1( a ) = 1, 1( b ) = 1, 1( c ) = 0 ; M1={a,b} and2( a ) = 1, 2( b ) = 0, 2( c ) = 1; M2={a,c}. Then (1,M1) and (2,M2) are gsfGsclosed but (1,M1)  (2,M2) is not gsfGs-closed in (X, T). Remark : 1.4.3

Example : 1.4.4 Let X = {a, b, c} and T = T = {(0,ϕ), (1,X), (λ,M)} where ( a ) = 1, ( b ) = 0, ( c ) = 0; M={a}. Define fuzzy sets 1 and 2 as follows: 1( a ) = 0, 1( b ) = 0, 1( c ) = 1 ; M1={a,b} ,2( a ) = 0, 2( b ) = 1, 2( c ) = 0, M2={b}. Then (1,M1) and (2,M2) are gsfFs-open but (1,M1)  (2,M2) is not gsfFsopen in (X, T). Property : 1.4.2 Let (X, T) be any soft fuzzy topological space. A soft fuzzy set (,M) in (X, T) is gsfFsopen if and only if (,N) ≤ sfsint ( ,M ) whenever (,N) is soft fuzzy F in (X, T) and (,N) ≤(,M). Proof : Assume that (,M) is a gsfFs-open set in (X, T). Let (,N) be soft fuzzy F in (X, T) such that (,N) ≤ (,M). Then, (1,X)(,N) is soft fuzzy G and (1,X)(,M)(1,X)(,N). Since (1,X)(,M) is gsfGs-closed, sfscl (1,X)(,M) ≤ (1,X)(,N). That is, (1,X)sfsint ( ,M ) ≤ (1,X)(,N). Hence (,N) ≤ sfsint ( ,M ).Conversely, suppose that (,M) is a soft fuzzy set such that (,N) ≤ sfsint ( ,M ) whenever (,N) is soft fuzzy F and (,N) ≤ (,M). Let (1,X)(,M) ≤ (,F), (µ,F) is a soft fuzzy G set. Then, (1,X)(,F) ≤ (,M). By assumption,(1,X)(,M) ≤ sfsint ( ,M ). That is, (1,X)sfsint ( ,M ) ≤ (,F). Which implies sfscl ( (1,X)(,M) ) (,F). Thus (1,X)(,M) is a gsfGsclosed set in (X, T). Hence (,M) is a gsfFs-open set in (X, T). Property : 1.4.3 Let (X,T) be any soft fuzzy topological space. Let (1,M1), (2,M2)be soft fuzzy sets in a soft fuzzy topological space (X, T). Then the following hold : (i)

If (1,M1)is gsfGs-closed in (X, T) and (1,M1)(2,M2)sfscl ( 1,M1), then (2,M2) is gsfGs-closed.

The union of any two gsfFs-open sets need not be gsfFs-open as seen in the following example. Sri Sarada International Journal of Multidisciplinary Research Vol : 1, Issue: 1, Jan 2018

Page 13

If (1,M1)is gsfFs-open in (X, T) and sfsint ( 1,M1) (2,M2)(1,M1), then (2,M2) is gsfFs-open. Property : 1.4.4

Example : 1.5.1

Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. If (,M) is a gsfGs-closed set in (X, T) and if f : (X, T)  (Y, S) is soft fuzzy G irresolute and strongly sfs-closed, then f ( ,M) is gsfGs-closed in (Y, S).

 x X, M={0,1} and Y = {a}, S={(0,ϕ),(1,Y), (,Y)} where ( a ) = 0.62. Let f : (X, T)  (Y, S) be the identity function. Thenf is gsfs-continuous but not

(ii)

Proof : Let (,N) be any soft fuzzy G set in (Y, S) such that

1

f ( ,M ) (,N). Then (,M)f ( 1

,N ). Since (,M) is gsfGs-closed and f ( ,N ) is 1

soft fuzzy G, sfscl ( ,M ) f ( ,N ) . That is, f ( sfscl ( ,M )) (,N).

Let X = [0, 1], T = {(0,ϕ), (1,X), (n,M)} n( x ) =

where

1

gsfGs-continuous. Since f ((1,Y)(,Y)) is not gsfGs-closed in (X, T) for the soft fuzzy closed set ((1,Y)(,Y)) in (Y, S). Property : 1.5.2 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. For a function f : (X, T)  (Y, S), the following statements are equivalent : (a) (b)

(1.3.1)

Since f is strongly sfs-closed, f ( sfscl ( ,M )) is soft fuzzy closed in (Y, S). Now, f ( ,M ) f ( sfscl ( ,M )). Thus, sfscl ( f ( ,M )) sfscl (f( sfscl ( ,M ))). Since f ( sfscl ( λ,M )) is soft fuzzy closed in (Y, S), sfscl ( f ( ,M )) f ( sfscl ( ,M )). By (1.3.1), sfscl ( f ( ,M )) (,N). Hence f ( ,M ) is gsfGs-closed in (Y, S). 1.5 : GENERALIZATIONS OF SOFT FUZZY G SEMI-CONTINUOUS FUNCTIONS Definition : 1.5.1

2n  3 (n = 1, 2, 3, ......), 5n  1

f is gsfGs-continuous. The inverse image of every soft fuzzy open set in (Y, S) is gsfFsopen in (X, T).

Property : 1.5.3 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. If f : (X,T)  (Y,S) is gsfGscontinuous, then f ( gsfGs-cl ( ,M )) sfcl ( f ( ,M )) for any fuzzy set(,M) in (X, T). Proof : Let (,M) be any fuzzy set in (X, T). Then, 1

1

Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. A function f : (X, T)  (Y, S) is called generalized soft fuzzy G semi-continuous (in short, gsfGs-continuous) if the inverse image of every soft fuzzy closed set in (Y,S) is gsfGsclosed in (X, T).

(,M)f ( f ( ,M )) f ( sfcl ( f ( ,M ))). Thus,

Property : 1.5.1

gsfGs-cl ( ,M ) f ( sfcl ( f ( ,M ))). Hence, f ( gsfGs-cl ( ,M )) sfcl ( f ( ,M )).

Let (X, T) and (Y, S) be any two soft fuzzy topological spaces and f : (X, T)  (Y, S) be gsfGscontinuous then f is gsfs-continuous. Remark : 1.5.1 Every gsfs-continuous function need not be gsfGscontinuous as shown in the following example.

1

(,M)f

( sfcl ( f ( ,M ))). Since f is gsfGs1

continuous, f ( sfcl ( f ( ,M ))) is gsfGsclosed in (X, T). Now, gsfG s-cl ( ,M )  1

1

gsfGs-cl ( f ( sfcl ( f ( ,M )))). Since f ( sfcl ( f ( ,M ))) is gsfGs-closed in (X, T), 1

Property : 1.5.4 Let (X, T), (Y, S) and (Z, R) be any three soft fuzzy topological spaces. If f : (X, T)  (Y, S) is gsfGs-continuous and g : (Y, S)  (Z, R) is soft


Page 14

fuzzy continuous, then the composition gof : (X, T)  (Z, R) is gsfGs-continuous. Remark : 1.5.2 Composition of two gsfGs-continuous functions need not be gsfGs-continuous as shown in the following example.

1

Example : 1.5.2 Let X = [0, 1], T = {(0,ϕ), (1,X), (λn,N)} where

n( x ) =

2n  3 (n = 1, 2, 3, 5n  1

...... ),  x  X, N={0.5} . Y = {p}, S = {(0,ϕ), (1,Y),(1,Y), (µ2,Y)} where 1( p ) = 0.75, µ2( p ) = 0.52. Z = {p}, R = {(0,ϕ), (1,Z),(δ,Z)} where δ( p ) = 0.62. Let f : (X, T)  (Y, S) and g : (Y, S)  (Z, R) be identity functions. Then f and gare gsfGscontinuous, but the composition is not gsfGscontinuous. Since (gof)1 ( (1,Z)(,Z) ) is not gsfGs-closed in (X, T) for the soft fuzzy closed set ((1,Z)(,Z) ) in (Z, R). Definition : 1.5.2 A soft fuzzy topological space (X, T) is called soft fuzzy T-F space if every gsfFs-open set is soft fuzzy open. Equivalently every gsfGsclosed set is soft fuzzy closed. Property : 1.5.5 Let (X, T), (Y, S) and (Z, R) be any three soft fuzzy topological spaces and f : (X, T)  (Y, S) be soft fuzzy continuous and g : (Y, S)  (Z, R) be gsfGs-continuous. If (Y, S) is soft fuzzy T-F space, then the composition gof : (X, T)  ( Z, R) is soft fuzzy continuous. Remark : 1.5.3 The above Property : 1.5.5 is not valid if (Y, S) is not soft fuzzy T-Fas shown in the following example. Example : 1.5.3

Let X ={a} and T={(0,ϕ), (1,X), (1,X), (2,X), (3,X)} where 1( a ) = 0.67, 2( a ) = 0.20, 3( a ) = 0.50. Y ={p} and S = {(0,ϕ), (1,Y), (1,Y), (2,Y)} where 1( p ) = 0.67, 2( p ) = 0.20. Z = {p} and R = {(0,ϕ), (1,Z),(,Z)} where ( p ) = 0.40. Let f : (X, T)  (Y, S) and g : (Y, S)  (Z, R) be the identity functions. Then fis soft fuzzy continuous andgis gsfGscontinuous, but the compositiongof is not soft fuzzy continuous. Since (gof) ((1,Z),Z)) is not soft fuzzy closed in (X, T), for the soft fuzzy closed set ((1,Z),Z)) in (Z, R). Property : 1.5.6 Let (X, T), (Y, S) and (Z, R) be any three soft fuzzy topological spaces and f : (X, T)  (Y, S) be gsfs-continuous and g : (Y, S)  (Z, R) be gsfGs-continuous. If (Y, S) is soft fuzzy T-F space, then the composition gof : (X, T)  ( Z, R) is gsfscontinuous. Remark : 1.5.4 The above Property : 1.5.6 is not valid if (Y, S) is not soft fuzzy T-Fas shown in the following example. Example : 1.5.4 Let X ={a, b, c} and T1 ={(0,ϕ), (1,X), (1,N)}, T2 = {((0,ϕ), (1,X),(2,M), (3,P)}, T3 ={(0,ϕ), (1,X),(4,R)} where 1( a ) =1,1( b )=1, 1( c )= 0, N={a, b}; λ2( a )= 0, λ2( b ) = 1, λ2( c ) = 1, M={b, c}; λ3( a )= 1, λ3( b ) = 0, λ3( c ) = 0, P={a}; λ4( a ) = 1,λ4( b ) = 0, λ4( c ) = 1, R={a,c}. Let f: (X, T1)  (X, T2) be a function defined by f ( a )= f ( c )=c, f ( b ) = b and g : (X, T2)  (X, T3) be the identity function. Thenfis gsfs-continuous andgis gsfGs-continuous. But the composition gof is not 1

gsfs-continuous. Since (gof) ((1,X)(4,R)) is not gsfs-closed in (X, T1), for the soft fuzzy closed set ((1,X)(4,R)) in(X, T3). Definition : 1.5.3 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. A function f : (X, T)  (Y, S) is called gsfGs-irresolute if the inverse


Page 15

image of every gsfGs-closed set in (Y, S) is gsfGsclosed in (X, T). Property : 1.5.7 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. If f : (X, T)  (Y, S) is gsfGsirresolute then f is gsfGs-continuous. Remark : 1.5.5 Every gsfGs-continuous function need not be gsfGs-irresolute as shown in the following example. Example : 1.5.5 Let X = [0, 1], T = {(0,ϕ), (1,X), (λn,N)} where

n( x ) =

2n  3 (n = 1, 2, 3, ...... 5n  1

),  x  X, N={1}. Y = {p}, S = {(0,ϕ), (1,Y), (µ1,Y), (µ2,Y)} where 1( p ) = 0.64and µ2( p ) = 0.52. Let f : (X, T)  (Y, S) be the identity function. Then f isgsfGs-continuous but not gsfGsirresolute, for the soft fuzzy set (λ,Y) defined by λ( 1

p ) = 0.38 which is gsfGs-closed in (Y, S) butf λ,Y ) is not gsfGs-closed in (X, T).

(

Property : 1.5.8 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. For a function f : (X, T)  (Y, S), the following statements are equivalent : (a) f is gsfGs-irresolute.

Let (X, T), (Y, S) and (Z, R) be any three soft fuzzy topological spaces and f : (X, T)  (Y, S), g:(Y, S)  (Z, R) be functions. (i)

If f and g are gsfGs-irresolute, then the composition gof is gsfGs-irresolute.

(ii)

If f is gsfGs-irresolute and g is gsfGscontinuous then the composition gof is gsfGs-continuous.

Definition : 1.5.4 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. A function f : (X, T)  (Y, S) is said to be strongly gsfGs-continuous if the inverse image of every gsfGs-closed set in (Y, S) is soft fuzzy closed in (X, T). Definition : 1.5.5 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. A function f : (X, T)  (Y, S) is said to be perfectly gsfGs-continuous if the inverse image of every gsfGs-closed set in (Y, S) is both soft fuzzy open and soft fuzzy closed in (X, T). Property : 1.5.11 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces and f : (X, T)  (Y, S) be strongly gsfs-continuous then f is strongly gsfGscontinuous. Remark : 1.5.6

(b) The inverse image of every gsfFs-open set in (Y, S) is gsfFs-open in (X, T).

Strongly gsfGs-continuous function need not be strongly gsfs-continuous as shown in the following example.

Property : 1.5.9

Example : 1.5.6

Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. If f : (X, T)(Y, S) is gsfGsirresolute, then f ( gsfGs-cl ( ,M ))  gsfGs-cl ( f ( ,M )) for all (,M) in (X, T).

Let X = [0, 1], T1 = {(0,ϕ), (1,X), (λ,M)} where 0.64 ≤ λ(x) ≤ 1, M={1} and T2 = {(0,ϕ),

Property : 1.5.10

),  x  X, N={1}. Let f : (X, T1)  (X, T2) be the identity function. Then f is strongly gsfGscontinuous but not strongly gsfs-continuous, for the soft fuzzy set (µ,L) defined by µ ( x ) = 0.55,

(1,X), (λn,N)} where n( x ) =


2n  3 (n = 2, 3, ...... 5n  1

Page 16

1

L={0,1} which is gsfs-closed in (X, T2) but f ( µ,L ) is not soft fuzzy closed in (X, T1). Property : 1.5.12 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. A function f : (X, T)  (Y, S) is strongly gsfGs-continuous if and only if the inverse image of every gsfFs-open set in (Y, S) is soft fuzzy open in (X, T). Property : 1.5.13 Let (X, T), (Y, S) and (Z, R) be any three soft fuzzy topological spaces and f : (X, T)  (Y, S) , g : (Y, S)  (Z, R) be the functions. (i) If f is strongly gsfGs-continuous, g is gsfGs-continuous then gof is soft fuzzy continuous. (ii) If fis strongly gsfGs-continuous, g is gsfGs-irresolute then gof is strongly gsfGs-continuous. Property : 1.5.14

Property : 1.5.15 Let (X, T) and (Y, S) be any two soft fuzzy topolocical spaces. A function f : (X, T)  (Y, S) is perfectly gsfGs-continuous if and only if the inverse image of every gsfF s-open set in (Y, S) is both soft fuzzy open and soft fuzzy closed in (X, T). Property : 1.5.16 Let (X, T), (Y, S) and (Z, R) be any three soft fuzzy topological spaces and f : (X, T)  (Y, S),g : (Y, S) (Z, R) be the functions. If fis perfectly gsfGscontinuous, g is gsfGs-irresolute then gof is perfectly gsfGs-continuous. Remark : 1.5.8 From the results proved above, the following implications are obtained : Perfectly gsfs continuous

Strongly gsfs continuous

Let (X, T) and (Y, S) be any two soft fuzzy topological spaces and f : (X, T)  (Y, S). If f is perfectly gsfs-continuous then f is perfectly gsfGs -continuous.

Strongly gsfGs- continuous

(i) If f is perfectly gsfGs-continuous then f is strongly gsfGs-continuous. Remark : 1.5.7 Every strongly gsfGs-continuous function need not be perfectly gsfGs-continuous as shown in the following example. Example : 1.5.7 Let X = {a, b, c}, T = {(0,ϕ), (1,X), (λ,M)} and Y = {p, q, r} , S = {(0,ϕ), (1,Y), (µ1,N1), (µ2,N2)} where λ( a ) = 0, λ( b ) = 0, λ( c ) = 1, M={c}; µ1( p ) = 0, µ1( q ) = 0, µ1( r ) = 1, N1={r}; µ2( p ) = 1, µ2( q ) = 1, µ2( r ) = 0, N2={p,q}. Let f : (X, T)  (Y, S) be defined by f ( a ) = p,f ( b ) = p, f ( c ) = q. Thenfis strongly gsfGs-continuous but not perfectly gsfGscontinuous.

Perfectly gsfGs- continuous 1.6 :GENERALIZED CONNECTED SPACE

SOFT

FUZZY

GSEMI-

Definition : 1.6.1 A soft fuzzy topological space (X, T) is said to be generalized soft fuzzy G semi connected (in short, gsfGs-connected) if the only soft fuzzy sets which are both gsfGs-closed and gsfFs-open are (0,ϕ) and (1,X). Example : 1.6.1 Let X = [0, 1] and T = {(0,ϕ), (1,X), (λn,M)} where

n( x ) =

5n 2 (n = 2, 3, ....), x 7(n 2  1)


Page 17

 X, M={0,1}. There exists no proper soft fuzzy set in (X, T)which is both gsfGs-closed and gsfFsopen. Hence (X, T) is gsfGs-connected. Property : 1.6.1 Every gsfs-connected space is gsfGsconnected. Proof : Let (X, T) be a gsfs-connected space and suppose that (X, T) is not gsfGs-connected. Then there exists a proper soft fuzzy set (,M) of (X, T) which is both gsfFs-open and gsfGs-closed. Since every gsfFs-open set is gsfs-open and every gsfGs-closed set is gsfs-closed, (,M) is both gsfsclosed and gsfs-open. Hence (X, T) is not gsfsconnected. Contradiction. Hence every gsfsconnected space is gsfGs-connected. Property : 1.6.2 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces and f :(X,T)  (Y,S) (i) If f is gsfGs-continuous surjection and (X, T) is gsfGs-connected, then (Y, S) is soft fuzzy connected. (ii) If f is gsfGs-irresolute surjection and (X, T) is gsfGs-connected, then (Y, S) is gsfGs-connected. (iii) If f is strongly gsfGs-continuous surjection and (X, T) is soft fuzzy connected, then (Y, S) is gsfGs-connected. Property : 1.6.3 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. If f : (X, T)  (Y, S) is gsfGsirresolute surjection, (X, T) is soft fuzzy T-Fσ and soft fuzzy connected, then (Y, S) is gsfGsconnected. Proof : Suppose that (Y, S) is not gsfGs-connected. Then there exists proper soft fuzzy set (,M) which is both gsfGs-closed and gsfFs-open in (Y, S). Since f is gsfGs-irresolute surjection,

1

f

both gsfGs-closed and gsfFs-open in (X, T). Since 1

(X, T) is soft fuzzy T-Fσ , f ( ,M ) is both soft fuzzy closed and soft fuzzy open in (X, T). Thus (X, T) is not soft fuzzy connected. Contradiction. Hence (Y, S) is gsfGs-connected. 1.7: GENERALIZED SOFT FUZZY G SEMIEXTREMALLY DISCONNECTED SPACE AND GENERALIZED SOFT FUZZY FSEMI-COMPACT SPACE Definition : 1.7.1 A soft fuzzy topological space (X, T) is said to be generalized soft fuzzy G semi extremally disconnected (in short, gsfGs-extremally disconnected) if gsfGs-cl (,M) is gsfFs-open, whenever (,M) is gsfFs-open. Property : 1.7.1 Let (X, T) be any soft fuzzy topological space and (X, T) be gsfGs-extremally disconnected space. Then the following statements are hold. (i)

For each gsfGs-closed set (,M), gsfFsint ( ,M ) is gsfGs-closed. (ii) For each gsfFs-open set (,M). gsfGs-cl(,M)+gsfGs-cl((1,X)gsfGs-cl(,M))= (1,X). Proof : (i) Let (,M) be any gsfGs-closed set in (X, T). Then (1,X)(,M) is gsfFs-open in (X, T). Since (X, T) is gsfGs-extremally disconnected space, gsfGscl ((1,X)(,M)) is gsfFs-open. Now, gsfGs-cl ( (1,X)(,M) ) = (1,X)gsfFs-int ( ,M ) is gsfFsopen . Hence gsfFs-int (,M) is gsfGs-closed. (ii) Let (,M) be any gsfFs-open set in (X, T). Since (1,X)gsfGs-cl ( ,M ) = gsfFs-int ( (1,X)(,M) ), gsfGs-cl ((1,X) gsfGs - cl ( ,M ) ) = gsfGs - cl ( gsfFs -int ((1,X)(,M)). Thus,

( ,M) is


Page 18

gsfGs-cl ( ,M ) + gsfGs-cl ( (1,X)gsfGs-cl ( ,M ))= gsfGs-cl ( ,M ((1,X)(,M)).

)

+ gsfGs-cl (

gsfFs-int

Since (,M) is gsfFs-open, (1,X)(,M) is gsfGsclosed. By (i), gsfF s-int ((1,X)(,M)) is gsfGs-closed. Hence, gsfGs-cl(gsfFs-int ((1,X)(,M)))=gsfFs-int ((1,X)(,M)).

(1.6.4)

iJ

Since f is gsfGs-continuous bijective function and each (i,Mi)is soft fuzzy open in (Y, S), f -1(i,Mi) is gsfFs-open in (X, T). From (1.6.4), (1,X)  f

-

iJ

1( ,M ). i i

Now, { f -1(i,Mi)} iJ is a gsfFs-open cover of (X, T). Since (X, T) is gsfFs-compact, there exists a finite subset F of J such that (1,X)  f iF

1( ,M ). i i

Then, f (1,X)   (i,Mi).That is, (1,Y)  iF

Therefore,

iF

(i,Mi). Hence (Y, S) is soft fuzzy compact.

gsfGs-cl ( ,M ) + gsfGs-cl ((1,X)gsfGs-cl ( ,M )) = gsfGs-cl ((1,X)(,M)).

(1,Y)  {(i,Mi)}

(,M)

+

gsfFs-int

= gsfGs-cl (,M) + (1,X) gsfGs-cl ( ,M ). = (1,X). Hence, gsfGs-cl( ,M ) + gsfGs-cl ((1,X)gsfGs-cl ( ,M )) =(1,X). Definition : 1.7.2 A collection {(i,Mi)}iJ of soft fuzzy sets of a soft fuzzy topological space (X, T) is called gsfFs- open cover of (X, T), if (i,Mi)s (iJ) are gsfFs-open sets of (X, T). Definition : 1.7.3 A soft fuzzy topological space (X, T) is called gsfFs-compact if every gsfFs-open cover of (X, T) has a finite subcover. Property : 1.7.2 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces. If f : (X, T)  (Y, S) is gsfGscontinuous bijective function and (X, T) is gsfFscompact, then (Y, S) is soft fuzzy compact. Proof : Let {(i,Mi)}iJ be a collection of soft fuzzy open sets in (Y, S) such that

Property : 1.7.3 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces.If f : (X, T)  (Y, S) is gsfGsirresolute bijective function and (X, T) is gsfF scompact, then (Y, S) is gsfFs-compact. Property : 1.7.4 Let (X, T) and (Y, S) be any two soft fuzzy topological spaces.If f : (X, T)  (Y, S) is strongly gsfGs-continuous bijective function and (X, T) is soft fuzzy compact, then (Y, S) is gsfFs-compact.

REFERENCES: [1] K. K. Azad, On fuzzy semi continuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl., 82 ( 1981 ), 14 - 32. [2] G. Balasubramanian, Maximal fuzzy topologies, KYBERNETIKA, 31 (1995 ), 459 – 464. [3] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 ( 1968 ), 182 - 190. [4] U. V.Fatteh and D. S. Bassan, Fuzzy connectedness and its stronger forms, J. Math. Anal. Appl., 111 ( 1985 ), 449 - 464. [5] Jin Han Park and Jin Kuen Park, Generalized fuzzy semi-closed sets and generalizations of fuzzy continuousfunction, International Reviewof Fuzzy Mathematics, Volume 1, No. 1 2006, pp. 89 - 101. [6] P. Smets, The degree of belief in a fuzzy event, Inform. Sci., 25(1981) 1-19. [7] M. Sugeno, An introductory survey of fuzzy control, Inform. Sci., 36(1985) 59-83. [8] Ismail. U. Tiryaki, Fuzzy sets over the poset I, Hacet. J. Math. Stat., 37(2)(2008) 143-166. [9] V. Visalakshi, M. K. Uma, E. Roja: On soft fuzzy Gδ pre continuity in soft fuzzy topological space, Annals of Fuzzy Mathematics and Informatics, Vol. 8, No. 6(2014), 921-939. [10] L. A. Zadeh, Fuzzy sets, Information and Control, 8(1965) 338-353.


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STRUCTURAL AND OPTICAL CHARACTERISTICS OF PURE AND CADMIUM DOPED COPPER OXIDE NANOPARTICLES R.Esaivani, Dr. S.S.Gomathi, P.Monisha and Dr.G.K. Vanathi Nachiyar Department of Physics, Sri Sarada College for Women (Autonomous), Salem – 636 016, India [email protected]

Abstract: Pure and cadmium – doped copper oxide nanoparticles have been prepared by chemical precipitation method using copper (II) sulphatepentahydrateand cadmium acetatedihydrate as precursors with NaOH as stabilizing agent. The synthesized pure and cadmium doped copper oxide nanoparticles were examined by X – ray Diffractometer, Ultraviolet - Visible Spectrometer, Fourier Transform Infrared Spectrometer, Photoluminescence Spectrometer, Scanning Electron Microscope and Energy Dispersive X – ray Spectrometer. The grain sizes were calculated by XRD. The optical properties were studied by UV – Visible Spectroscopy. The functional groups and chemical interactions of samples were also determined at various peaks using FTIR data. The surface morphology of the prepared nanoparticles was analysed by SEM. The presence of dopant in the doped samples wasconfirmed using EDAX measurements. 1. INTRODUCTION Nanoparticles of metal oxides are manufactured in the large scale because of their industrial applications. Copper oxide which is an important p-type semiconductor, has drawn increasing attention in the applications of gas sensors because of low cost, excellent reactivity, high stability, non-toxicity and also secondary architectures composed of nanostructured building blocks have attracted significant interest in material and device fabrications. Wide bandgap oxide semiconductors, when doped with

transition metal ions have attracted much attention for their promising versatile applications. CuO with narrow band gap of 1.2 eV is extensively used in various applications such as catalysis [1], solar energy conversion [2], gas sensor [3] and field emission [4]. However, these novel properties can be improved by synthesis in CuO nanostructures that shown excellent performance comparing to bulk counterpart. Different nanostructuresof CuO are synthesized in the form of nanowire, nanorod, nanoneedle, nanoflower and nanoparticle. In thepast decades, various methods have been proposed to produce CuO nanoparticles with different sizes andshapes such as thermal oxidation [5], sonochemical [6], combustion [7] and quick-precipitation [89].Among these processes, precipitation method is a facile way which attracts considerable interest inindustries because of low energy and temperature, inexpensive and cost-effective approach for large scaleproduction and good yield. The main objective of the present work is to investigate the effect of cadmium on the structural and optical properties of CuO nanostructures synthesized via precipitation method.The morphology and composition of the synthesized samples were studied using scanning electron microscopy and EDAX. 2. MATERIALS AND METHODS To synthesize copper oxide nanoparticles, Copper (II) sulphate pentahydrate (CuSO4.5H2O), cadmium acetate dihydrate (Cd(CH3COO2).2H2O) and sodium hydroxide (NaOH) were used. Analytical reagent grade chemicals were used in the synthesis and the glasswares used in this work were acetone washed. Double distilled water and ethanol were used as solvent for sample preparation. During the synthesis of CuO-NPs, 5g of copper (II) sulphate (CuSO4.5H2O) and sodium hydroxide pellets were dissolved in double distilled water. Drop wise addition of sodium hydroxide solution to copper (II) sulphate pentahydrate solution was carried out with constant stirring at room temperature. After 3 hours of stirring, the solution was taken out from the stirrer and kept aside. The


Page 20

blue precipitate appeared quickly indicating the formation of pure CuO. Finally, the precipitate was rinsed with distilled water many times by adding ethanol to remove impurities until pH of 78 was reached. Then the precipitate was filtered using filter paper. After that, the precipitate was dried in hot air oven by maintaining it at the temperature of 100C. Then the sample was grinded to get the powder nanoparticles. Schematically the chemical reaction can be represented as: CuSO4 .5H2O+ 2NaOH → Cu(OH)2+ Na2SO4 +5H2O And copper hydroxide decomposes into copper oxide on heating as follows, Cu(OH)2 CuO +H2O For the doped samples the same procedure was repeated but different concentration of (3%)Cd(CH3COO)2.2H2O was added to copper sulphate solution and the rest of the procedure wasrepeated as puresample preparation. The crystal of the synthesized nanoparticles were studied by powder X-ray diffraction method (XRD) using a PhilipsPW-1710 X-Ray Diffractometer with CuKα radiation (λ = 1.54056 Å) in the 2θ range of 10-80°at room temperature. Theabsorption and transmittance properties of the samples were studied by UltravioletVisible(UV-Vis) Spectrometer (Perkin Elmer, Lambda 35) in the wavelength range from 190 to 1100nm at room temperature.To identify the functional groups and toconfirm the substitution of Cd ions, the samples were examined with Fourier TransformInfrared Spectrometer (FTIR; RX1 PERKINELMER: USA) at a resolution of 2 cm-1. Themeasurements were carried out in the region 400 – 4000 cm-1 using KBr as the beam splitter.The morphology of the samples was obtained using ScanningElectron Microscope(Jeol JSM 6390) with Energy Dispersive Analysis of XRay(EDAX) at an accelerating voltage of 200 kV. Photoluminescence (PL) Spectrum was recorded at roomtemperature with a He-Cd laser line at 405 nm used as an excitation wavelength by means

ofspectrophotometer (Kimon, SPEC-14031 K, Japan). 3. RESULTS AND DISCUSSION 3.1 X- Ray Diffraction Analysis The X-ray diffraction patterns (XRD) of pure and 3% Cd doped CuO nanoparticles are shown in Fig.1.(aand b).All peaks are well indexed to the monoclinic phase of CuO that was confirmed from JCPDS card No. 05-0661. The characteristic peaks located at 2θ = 32.22:,35.40:, 38.50:, 48.90:, 53.40:, 57.20:, 61.50:, 65.80:, 67.30:, 74.60:and 82.20:are assigned to (110), (002), (111), (-202), (020), (202), (-113), (-311), (220), (-222) and (222) plane orientation of CuO (JCPDS 05-0661). The main peaks at 2θ= 35.4: and 38.48:corresponding to (002) and (111) planes are the characteristics peaks for monoclinic phase of pure CuO nanoparticles. Further no other impurity peak was observed in the XRD pattern showing the single

phase sample formation of CuO nanoparticles. Fig.1. XRD patterns of a)Pure CuO b)3% Cd doped CuO Nanoparticles Matching the XRD patterns of the reported data with the CuO samples, results showthat the Cd doped CuO is well crystalline and the predominant diffraction lines are same as that of pure CuO. The crystallite size (D) was calculated using DebyeScherrer’s formula

D

0.9  cos 

where λ is the wavelength of Cu-Kα radiation, β is the full-width at half maximum (FWHM) and θ is the diffraction angle. The crystallite sizes were


Page 21

calculated using XRD data and are presented in table 1. The crystallite size increases with dopant concentration of 3%. Table 1. Crystallite size calculation from XRD profile Posi tion of Avera Interpl Crys ge 2of anar tallit (hkl) Samp Cryst the distanc e valu le allite inte e ‘d’ Size( es size( nse (Å) nm) nm) peak (Deg ) 38.4 2.3371 11.2 111 Pure 883 2 2 11.75 CuO 35.3 2.5342 12.2 002 908 4 7 38.8 2.3185 11.9 111 3%Cd 093 2 0 dope 12.37 35.5 2.5202 12.8 d CuO 002 938 0 6 The average crystallite size of pure CuO is 11.75 nm and that for Cd doped CuO is 12.37 nm. The addition of cadmium causes a slight increase in the size of the nanoparticles. In the case of pure CuO spectrum the diffraction peaks are considerably broadened that is attributed to the small crystallite sizes. Small crystallites have relatively few lattice planes that contribute to the diffraction lines. Broadening of the peaks may occur due to micro straining of the crystal structure arising from defects and the defects are negligible in the case of Cd doped CuO. The diffraction peaks are considerably narrowed and the intensity is enhanced in the case of Cd doped CuO that are attributed to the increase in crystallite sizes [10-12]. 3.2 Optical Absorbance and Transmittance Study Absorption spectra of pure and Cd doped CuO nanoparticles were recorded using Perkin Elmer UV-Visible spectrophotometer. From this the band gap and the type of electronic transitions were

determined.Fig.2.shows the UV –Visible absorption spectra of the as-prepared pure CuO and Cd doped CuO nanostructures in the range of wavelength between 190 nm to 1100 nm at room temperature. All the samples exhibit broad absorption peaks extending from UV to visible range. It has been observed that the samples exhibit an absorption edge at around 245 nm. The absorption peak is at about 398.9 nm for pure CuO sample. With 3% Cd doping the optical absorption edge is slightly shifted towards longer wavelength of 647.6 nm which may be attributed to the increase in grain size.

Fig. 2. UV–Visible Absorbance Spectrum of a) PureCuOb) 3% Cd doped CuO Nanoparticles Optical band gap can be calculated from absorption spectrum using the relation

Eg 

1243



where 𝜆 is the excitonic absorption wavelength. Corresponding energy gap calculated for pure CuO and Cd doped CuO samples are 3.12eV and 1.92 eV respectively and is shown in table 2.

Table 2. Band Gap and Grain size calculation

Samples

Absorption Wavelength (nm)


Band gap(eV)

Grain size from XRD(n m)

Page 22

Pure CuO

398.8

3.12

11.75

3%Cd doped CuO

647.6

1.92

12.37

Fig. 4. FTIR Spectrum of a) Pure CuO b) 3% Cd Doped CuO Nanoparticles The band gap decreases initially with 3% Cd doping. This confirms that CuO crystals have semiconductor character, where they obtained direct band gap values depending on the size and the doping. The observed decrease in the direct band gap values of CuO nanoparticles with the increase in grain size is attributed to the quantumconfinement effect.[13,14,15] Optical transmittance spectra of the samples are displayed in Fig.3. The sample which is 3% Cd doped CuO prepared at room temperature has the higher transmittance of about 80% in the wavelength region between 200 nm and 1100 nm.The measurement shows that the transmittance of the 3% Cd doped CuO nanoparticles increases with the particle size and the maximum transmittance can be achieved with larger particles. The high transmittance of the prepared nanoparticles from ultraviolet to the Infrared (IR) region suggests that these particles could be employed as a host material for optoelectronic applications.

Metal oxides generally give absorption bands below 1000 cm-1that arise from inter atomic vibrations and by observing frequencies helps in the confirmation of particle formation. Fig.4. represents the FTIR spectra of pure and Cd doped CuO nanoparticles. It is clear that all the samples exhibit vibrations in the region 400 – 600 cm-1, which can be attributed to the vibrations of Cu(II)O bond. The broad absorption peak at around 3400 cm-1 is caused by the absorbed water molecules since the nanocrystalline materials exhibit a high surface to volume ratio and thus absorbs moisture. No impurity phase is recognized. Regarding functional groups, the band at ∼3400 cm−1 is related to the existence of hydroxyl groups (O-H). The FTIR spectrum of CuO shows three strong peaks at 494cm−1, 616 cm−1 and 418 cm−1 and Cd doped CuO at 421 cm−1, 608 cm−1and 409cm−1. There are associated with Cu-O vibrations of monoclinic CuO. The vibrational frequency of pure CuO at 878 cm−1 is shifted to 925 cm−1 in Cd doped CuO which confirms the presence of Cd-O stretching. There is a clear shift in the position of peaks of 3% Cd doped CuO nanoparticles which are related to quantum size and surface effects of nanomaterials [16]. 3.4 Photoluminescence Study Photoluminescence spectra of pure and Cd doped CuO nanoparticles recorded at room temperature is shown in Fig.5.

Fig. 5. Photoluminescence Spectrum of a) Pure CuO b) 3% Cd Doped CuO Nanoparticles

Fig.3. Optical Transmittance Spectrum of a) PureCuOb) 3% Cd doped CuO Nanoparticles

3.3. FTIR Spectroscopic Studies

The samples were excited using laser excitation source at a wavelength of 405 nm. The PL intensity of emission peak for 3% Cd doped CuO nanoparticle is enhanced compared to the


Page 23

intensity of emission peak for pure CuO nanoparticles. The PL spectrum of Cd doped CuO nanoparticles reveal one peak at 681 nm which is slightly greater than that of pure CuO which is at 680 nm. The luminescence properties of CuO are strongly dependent on the morphology and size of the formed nanostructures. Briefly, the PL properties of CuO nanostructures can be controlled by their size, shape and morphology. The two most reported mechanisms are quantum confinement effect and specific surface effect, which can result in the blue shift and red shift of the PL peak. Thus, considering these parameters in the design of CuO-based photodetection device is significant. Hence the strongest emission spectrum was observed for 3% Cd doped CuO nanoparticles at 681 nm which corresponds to red emission [17]. 3.5 Morphological Study Fig.6.shows the typical morphology of pure and 3% Cd doped CuO nanoparticles. SEM micrographs clearly exhibit almost polycrystalline, porous morphology with the nanofused surface. The needle rod like structure seen in the pure CuO is distorted to spherical nanoparticles in the case of Cd doped CuO nanoparticles which confirm that the microstructure of CuO is sensitive to Cd doping. The average particle size of pure CuO ranges from 24 nm to 48 nm while it was from 34 nm to 43 nm for 3% Cd doped CuO sample. The variation in particle size matches well with the XRD results. However, the average particle size obtained from SEM analysis is slightly greater than the values calculated from XRD measurements. It may be due to the aggregation of smaller particles during sample preparation for SEM analysis.

Fig.6. SEM Images of a) Pure CuOb) 3% Cd doped CuO Nanoparticles

3.6 Compositional Analysis The presence of Cd in doped samples is confirmed from the selective area EDAX analysis. EDAX spectra of pure CuO and 3% Cd doped CuO nanoparticles are shown in Fig. 7. It can be concluded that Cd is successfully doped in the CuO nanocrystals [12].

Fig.7.EDAX Spectrum of a) Pure CuO b) 3% Cd Doped CuO Nanoparticles

4. CONCLUSION Pure CuO and Cd doped CuO nanocrystalline powders have been synthesized by precipitation method. XRD patterns confirmed that the samples contain the nanoparticles of size 11.75 to 12.37 nm. The results of UV-Visible spectrum shows that the prepared pure CuO nanoparticles have absorption at 398.8 nm corresponding to the energy gap of 3.11 eV which is greater than the energy gap of bulk CuO (1.24 eV). The decrease or increase in the energy gap from 3.11 eV to 1.94 eV confirmed Cd doping. The energy gap of all the samples are greater than the energy gap of bulk CuO nanocrystals. The observed increase in the direct band gap values of CuO nanoparticles with the decrease in particle size is attributed to the quantum confinement effect. Further, the optical transmittance spectra show that Cd dopant increases the transmittance of CuO to nearly 80%. The maximum transmittance can be achieved with larger particles. The high transmittance of the prepared nanoparticles from ultraviolet to the red region suggests that these


Page 24

particles could be employed as a host material for optoelectronic applications. FTIR spectra confirmed the Cd substitution through the shift in the position of peaks of 3% Cd doped CuO nanoparticles which are related to quantum size and surface effects of nanomaterials. The PL studies confirmed the Cd substitution by means of emissions at much higher wavelengths of 680 nm and at 681 nm in the Red region. Therefore, PL results demonstrated that the as-synthesized high quality of CuO nanoparticles can be a promising candidate for industrially integrated optoelectronic applications. SEM micrographs revealed the Cd dopant through increase in particle size. In addition, SEM images confirmed that Cd is doped in CuO without disturbing the basic monoclinic structure of CuO. EDAX analysis also confirmed that the samples are composed of Cu, Cd and O with some impurity elements.

[11] Xiaogang Wen , Weixin Zhang and Shihe Yang, Synthesis of Cu(OH)2 and CuO nanoribbon arrays on a copper surface, Langmuir 19(14) 2003, 5898- 5903. [12]J.M. Wesselinowa, Magnetic properties of doped and undoped CuO nanoparticlestaking into account spinphonon interactions. Physics Letters A 375 (2011) 1417-1420. [13] J. Essic, R. Mather, Characterization of bulk semiconductors band gap via near- absorption edge optical transmission experiment. American Journal ofPhysics,Volume 61, Issue 7, (1993) 646-649. [14] R. Willardson, A. Beer, Optical Properties of IIIV Compounds. Academic Press NewYork,1967, pp 318400. [15] M. Dressel, G. Gruner, Optical and Electrical Properties of sol-gel synthesized Calcium Copper Titranate Nanopowders, Electrodynamics of Solid Optical Properties of Electron in Matter, Cambridge University Press 2009, 159-65. [16] J. T. Kloprogge, L. Hickey and R.L. Forst, FT-Raman and FT-IR Spectroscopic study of synthetic Mg/Zn/Al Hydrotalcites, Journal of Raman Spectroscopy 35(2004) 967-974. [17] V.V.T. Padk and M. Cernil, Green Synthesis of copper oxide Nanoparticles using Gum Karaya as a Biotemplate and their Antibacterial Application, International Journey of Nanomedicine 8(2013) 889-898.

REFERENCES [1] Jianliang C Yan, W Tianyi, Ma Yuping Liu, Zhongyong Yuan. Synthesis of porous hematitenanorods loaded with Cuo nanocrystals as catalysts for CO oxidation. J Nat Gas Chem 2011;20: 669-676. [2] Jess K, Nicolas G, Richard R, Eric Miller. Advances in copperchalcopyrite thin films for solarenergy conversion. Sol Energ Mat Sol C 2009; 94:12-16. [3] Yang Z, Xiuli He, Jianping L, Huigang Z, Xiaoguang G. Gassensing properties of hollow andhierarchical copper oxide microspheres. Sensor2007;128:293 298. [4] Bohr R H, Chun S Y, Dau C W, Tan J T, Sung J. Field emission studies of amorphous carbondeposited on copper nanowires grown by cathodic arc plasma deposition. New Carbon Mater 2009;24:97-101. [5] Manmeet K, Muthea K P, Despandeb S K, Shipra Ch, Singhd J B, Neetika V, Gupta S K, Yakhmi JV. Growth and branching of CuO nanowires by thermal oxidation of copper. J Cryst Growth2011;289:670-675. [6] Narongdet W, Piyanut C, Naratip V, Wisanu P. Sonochemical Synthesis andCharacterization ofCopper Oxide Nanoparticles. Energy Procedia 2011;29:404-409. [7] Yamukyan M H, Manukyan K V, Kharatyan S L. Copper oxide reduction by combinedreducersunder the combustion mode. Chem 2008;137:636 642. [8] J Zhu, D Li, H Chen, X Yang, L Lu, X Wang. Highly dispersed CuO nanoparticlesprepared by anovel quickprecipitation method. Mater Lett 2004;58:3324-3327. [9] Rujun W, Zhenye M, Zhenggui G, Yan Y. Preparation and characterization of CuOnanoparticleswith different morphology through a simple quick-precipitation method in DMAC water mixedsolvent. J Alloy Compd 2010;504:45 49. [10] Lizhi Zhang, Jimmy C. Yu, An-Wu Xu, Quan Li, Kwan Wai Kwong, Shu-Hong Yu Peanut –shaped nanoribbon bundle superstructures of malachite and copper oxide, Journal of Crystal Growth 266 (2004) 545-551.


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INVESTIGATION OF MOLECULAR STRUCTURE AND SPECTRAL ANALYSIS OF PROPYL SALICYLATE BASED ON DFT CALCULATIONS Dr.Mathammal.R, Dr.Hema Malini.R, Tamil Selvi.P, Mekala.R, Sangeetha.M Department of Physics, Sri Sarada College for Women (Autonomous), Salem-636 016, Tamil Nadu, India. [email protected].

ABSTRACT: This study reports the structural characterization of Propyl salicylate (PS) by using spectroscopic and quantum chemical methods. Propyl salicylate is used as an intermediate in manufacture of pesticides, drugs and perfumery products. Density functional method (DFT) has been used to calculate the optimized geometrical parameters, atomic charges, vibrational wavenumbers and intensity of the vibrational bands. The complete analysis of Propyl salicylate is performed by combining the experimental and theoretical information using density functional theory (DFT) employing B3LYP method with 6.31+ G(d,p). The Fourier Transform Infrared (FTIR) and FT-Raman of Propyl salicylate have been recorded and analyzed. The 1H and 13C NMR chemical shifts of the molecule were calculated by the Gauge Independent Atomic Orbital (GIAO) method and compared with experimental results. Natural Bond Orbital (NBO) analysis has been performed for analyzing charge delocalization throughout the molecule. Molecular electrostatic potential has also been used for quantitative measure of the chemical activities of various sites of the molecule. The theoretical UV-visible spectrum of the compound was performed by time-dependent DFT (TD-DFT) approach. Mulliken atomic charges of Propyl salicylate were calculated. Besides, charge transfer occuring in the molecule between HOMO and LUMO energies are calculated and presented. The non-linear optical properties were discussed from the dipole moment values and excitation wavelength in the UV-visible region.

Keywords: DFT, FTIR, FT-Raman, HOMO and LUMO, UV-visible 1. INTRODUCTION Viruses are the smallest known pathogens that are world widely spread.In present decades almost three quarters of contagious diseases in world are caused by viruses. Ester derivatives of hydroxy benzoic acid are reported to be widely used for treating infections caused by hepatitis B viruses, human papilloma, herpes simplex virus, cervicitis and cervical erosions in human and animals, and is also antioxidant, antibacterial, anti-inflammatory, antimutagenic and chemo preservative compound [1]. Many esters are known today which may either be natural or synthetic. Many of the important esters include the salicylates which are useful in the preservation of foods, pain control and fever control [2]. Salicylates are non-steroidal antiinflammatory agent [3].Salicylates are derivatives of salicylic acid that occur naturally in plants and serve as a natural immune hormone and preservative, protecting the plants against diseases, insects, fungi and harmful bacteria. Salicylates allow the formation of intramolecular hydrogen bonds, an effect which influences physical properties such as solubility and ionization when compared to the meta and paraisomers.Intramolecular hydrogen bonding stabilizes the anionic charge of the salicylic acid anion and facilities the ionization process, enhancing carboxylic acid strength [4]. Propyl alcohol is most commonly used as a solvent for waxes, vegetable oils, resins, cellulose esters and ethers [5]. Propyl salicylate (PS) is used as an intermediate in manufacture of pesticides, drugs and perfumery products [6]. The goal of the present study is to give a complete description of the molecular geometry and molecular vibrations of the title compound. The assignments of bands in the vibrational spectra of molecule are an essential step in the application of vibrational spectroscopy for solving various structural chemical problems. The complete analysis of Propyl salicylate is performed by combining the experimental and theoretical information using


Page 26

density functional theory (DFT) employing B3LYP exchange correlation and the 6-31+ G(d,p) basis set. The calculated infrared and Raman spectra of the title compound are also simulated utilizing the scaled force. NMR spectral analysis and Natural bond orbital (NBO) analysis have also been studied by using the same method and basis set. The theoretical UV-visible spectrum of the compound was performed by time-dependent DFT (TD-DFT) approach. The optimized geometric parameters were compared with available experimental data via the XRD results derived from complexes of this molecule. 2. EXPERIMENTAL STUDIES The compound Propyl salicylate was purchased from sigma-Aldrich chemical company (USA) with a stated purity of 97%. The FTIR spectrum of the title compound was recorded in the region 4004000 cm-1 using BRUCKER IFS 66V spectrophotometer provided with a Globar source, KBr beam splitter and MCT detector. The FT-Raman spectrum of the compound was recorded in the BRUCKER RFS 100/S FT-Raman spectrometer equipped with Nd:YAG laser source operating at 1064 nm line widths with 100 MW power, in the range of 4000-10 cm-1. The 1H and 13C NMR experiments were carried out on a BRUCKER AV III 400 NMR Spectrometer. 3. COMPUTATIONAL DETAILS All the calculations were performed using Density Functional Theory (DFT). 4. RESULTS AND DISCUSSION 4.1 MOLECULAR GEOMETRY The crystal structure of the studied molecule is not available; therefore, the optimized structure is compared with the structurally similar molecule propyl-para-hydroxybenzoate [7]. The optimized structure of the title compound with the global minimum energy is shown in the Fig 1.

Fig.1: The Optimized structure of PS The C-C bond lengths in the ring are measured at 1.384, 1.385, 1.374, 1.397, 1.386 and 1.375 Å for Propyl salicylate. The optimized C-C bond lengths in PS fall in the range 1.3872- 1.42 Å. These values show that our calculation results are more consistent with the experimental data. The Cring- C (carbonyl group) bond length was calculated 1.4717 Å and it shows good agreement with the experimental data of 1.476 Å. The increase in bond length for C2-C3 and C3-C4 are due to the presence of heavier group in the place of hydrogen. For title compound, the bond lengths of C=O and C-O are 1.213 Å, 1.328Å respectively and their calculated values are 1.2359 Å, 1.3425 Å. This result indicates that bond length decreases with increasing bond order.The average C-H bond length in the aromatic ring calculated by B3LYP/6-31 + G( d,p) is ~1.085 Å. It is less than C-H bond length present outside the ring. The structural analysis reveals the presence of intramolecular hydrogen bond interactions in PS between hydroxyl group and carbonyl group. In PS, the hydrogen bonded O11-H12 bond length is 0.9872 Å and coincides with the reported hydrogen bonded O-H distance (0.986) of 5-iodosalicylic acid [8]. 4.2 VIBRATIONAL ASSIGNMENTS Propyl salicylate molecule consists of 25 atoms, and therefore it has 69 normal modes of vibrations and it belongs to C1 point group. The fundamental vibrations of the title molecule can be distributed as 47A (in-plane) +22A (out of plane). All vibrations are active both in Raman and infrared absorption. The theoretical and experimental FTIR and FT-Raman spectra are shown in Figs.2 & 3.


Page 27

supplementary bands assigned at 672, 516, 418 cm-1 to C-C out-of-plane bending. These assignments are in line with the assignments proposed by the literature [9-11]. Hydroxyl group Fig.2: FTIR Spectrum of PS

Fig.3: FT-Raman Spectrum of PS C-H stretching The investigated molecule is di-substituted, therefore four aromatic C-H vibrations (C2-H8), (C1-H7), (C6-H10) and (C5-H9) are observed in vibrational spectra. The theoretically calculated scaled down vibrations at 3047, 3064, 3074 and 3084 cm-1 corresponding with C-H stretch show an excellent agreement with the experimentally observed vibrations at 3074 and 3076 cm-1 in FTIR and FT-Raman respectively. The calculated values for C-H out-of-plane bending vibrations are 733, 777, 840 and 883 cm-1. Its experimental counterparts are identified in IR at 846 and 864 cm-1 and in FT-Raman at 897 cm-1. The bands at 1068, 1129, 1132 and 1189 cm-1 are assigned to C-H in-plane bending vibrations and experimentally observed at 1090, 1137 and 1183 cm-1 in FTIR and at 1139 cm-1 in FT-Raman spectra. All these bands are in the expected range and also good agreement with literature values. C-C stretching The bands at 1583, 1553, 1454, 1429, 1368, 1310 cm-1 are assigned to C-C stretching modes and are in good agreement with experimental values. The two frequencies are slightly lower wavenumber compared to standard value due to steric effect.In the present work, three bands present at 774, 648 and 543 cm-1 assigned to C-C in-plane bending, which is in good agreement with FTIR value at 631 cm-1 and three

In this study, the O-H stretching mode of Propyl salicylate molecule are observed at 3200 cm-1 in FTIR spectrum and predicted at 3264 cm-1 (O11H12). The C-O stretching vibration in alcohols, produce a strong band near 1260-1000 cm-1 [12] and is sensitive to the nature of the substituent. Carbonyl group The C=O stretching vibration is observed at 1636 cm-1 and corresponding experimental FTIR value at 1616 cm-1 for the title molecule. The C-O stretching bands in ester are highly informative. [13]. The C-O stretching modes are computed at 1279 and 1061 cm-1 for the title molecule. This band observed at 1036 cm-1 in FTIR spectrum is assigned to C-O stretching vibration. The Propyl carbon attached to ester group has C-O stretching from 940-840 cm-1, where as in calculated method it is found to be 930cm-1 and observed at 939 cm-1 in IR spectrum. This is in good agreement from obtained value 733 cm-1. The bands in the region 540±80 cm-1 exhibit C-O out-of-plane deformation [14]. The calculated values at 508 and 427 cm-1 are assigned to C-O out-of-plane bending for the title molecule. Methylene group In the present study, the band resulting from the two methylene asymmetric stretch νas (CH2)1 and νs (CH2)2 are 3004 and 2952 cm-1, respectively for the title molecule. The two symmetric stretching at 2941 and 2903 cm-1 are predicted by B3LYP/631+ G(d,p) method. The bands at 2940 and 2900 cm-1 in FTIR spectrum and 2942 cm-1 in FTRaman spectrum are assigned to CH2 stretching vibrations. The scissoring CH2 vibrations are predicted at 1423 and 1425 cm-1 theoretically. Absorption of methylene twisting vibration observed in the 1350-1150 cm-1 region [12]. These bands are generally appreciably weaker.


Page 28

The bands at 1258 and 1237 cm-1 are calculated to twisting methylene groups. The value at 1262 cm1 is observed experimentally for both FTIR and FT-Raman spectrum. The CH2 rocking vibrations are assigned at 736 and 819 cm-1 which is in agreement with the literature values. Methyl group The CH3 asymmetric and symmetric stretching vibrations are predicted at 2981, 2971 cm-1 and 2905 cm-1 respectively and shows good agreement with experimental value at 2972 cm-1 in FTIR spectrum. The symmetrical bending vibration δs CH3) occurs near 1375 cm-1, the asymmetrical bending vibration δ as CH3) near 1450 cm-1. The asymmetric CH3 vibration at 1445 cm-1 and symmetric bending vibration at 1322 and 1351 cm-1 are calculated theoretically. The bands observed at 1327 and 1351 cm-1 IR spectrum 1326 cm-1 in Raman spectrum for the title molecule. The methyl rocks are expected in the ranges 1045 ± 75 and 970 ± 70 cm-1 [15] and in the present case these modes are assigned at 910 and 952 cm-1 theoretically (PS). This shows an excellent agreement with experimentally observed value at 952 cm-1 in the IR spectrum and at 917 cm-1 in the Raman spectrum. The methyl torsions are often assigned in the region 185 ± 65 cm-1. The calculated value of CH3 torsion vibration at 205cm-1. 4.3 NUCLEAR MAGNETIC RESONANCE The chemical shift value of 13C which is in the carbonyl group has been observed at 158.43 ppm. The experimental value obtained is about 170.27 ppm. Due to the influence of electronegative oxygen atom, low-field shift of C4 carbon is recorded at 151.22 ppm and calculated at 161.83 ppm in CDCL3 solution. The chemical shift of four carbon peaks in the ring attached to hydrogen atom are observed from 104.13 to 121.34 ppm are calculated from 117.59 to 135.55 ppm at B3LYP/6-31+ G(d,p) level of theory. The C16, C19 and C22 carbon atoms have smaller chemical shifts (both experimentally and theoretically) than the other carbon atoms, due to shielding affect which the non-electronegative property of

hydrogen atoms. The chemical shift of H21 atom is 11.75 ppm which has a higher value than the other hydrogen atoms, due to the strong hydrogen bonding. From experimental data, it seems that the methyl protons are observed at 1.02 ppm in CDCl3 solution due to electron releasing nature. The methylene Hydrogens on the carbon attached to the single- bonded oxygen are usually recorded in the range of 3.5 - 4.8 ppm and calculated at this range 4.28 ppm experimentally. The aromatic protons of organic molecules are usually recorded in the range of 7.00 - 8.00 ppm and ring protons are found in this region experimentally, that shows in the Table.1. The ring protons were calculated in the region of 6.84 to 7.83 ppm and observed at 7.12 to 8.38 ppm. The chemical shifts of aromatic protons of title molecule are higher than the other protons. Therefore, the electronic charge density around of these atoms can be effected the influence of rapid proton exchange, hydrogen bond, solvent effect etc., in the molecular system. TABLE 1: Experimental and theoretical 13C and 1H NMR isotropic chemical shifts (with respect to TMS and in CDCl3 solution) of PS compound (atom positions are numbered as in Fig.1)

Atoms C13

Theoretical Experi B3LYP/ mental 6-31+G(d,p) 170.27 158.43

Theoretical Experi B3LYP/ Atoms mental 6-31+G(d,p) H12 10.86 11.75

C4

161.83 151.22

H8

7.83

8.38

C6

135.55 121.34

H10

7.41

7.76

C2

129.90 117.22

H9

6.95

7.34

C5

117.59 105.01

H7

6.84

7.12

C1

119.07 104.13

H18

4.28

5.50

C3

112.71 99.81

H17

4.28

4.00

C16

66.89

58.50

H21

1.78

2.31

C19

22.05

16.73

H20

1.78

1.64

C22

10.44

2.99

H23

1.02

1.60

H25

1.02

1.26

H24

1.02

1.12


Page 29

4.4 FRONTIER MOLECULAR ORBITALS The important aspect of the frontier electron theory is the focus on the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO) [37]. The energy gap between HOMO and LUMO determine the kinetic stability, chemical reactivity and optical polarizability and chemical hardnesssoftness of a molecule [38,39]. The energy levels of the HOMO and LUMO orbitals computed at the B3LYP/ 6-31+ G(d,p) level for the title compound is represented in Fig.4. Fig.4 demonstrate that the energy values of HOMO and LUMO levels for PS are computed to be -0.23717 eV and -0.06200 eV respectively, and the energy difference is 0.17517 eV. The other important quantities such as electronegativity (χ), hardness (η), softness (ξ), and electrophilicity index (ψ) were deduced from ionization potential and electron affinity values. The value of electro negativity, chemical hardness, softness and electrophilicity index are 0.14958, 0.08758, 5.7087 and -0.087585 respectively, for the title molecule.

indicates the strongest repulsion. As can be seen from the MEP map of the title molecule (Fig 5), while regions having the positive potential are over the hydrogen atoms, the regions having the negative potential are over the electronegative oxygen atoms. These results show that the H atoms indicate the strongest attraction and oxygen atom indicates the strongest repulsion.

FIG.5: Molecular Electrostatic Potential Surface of PS

4.6 HYPERPOLARIZABILITY The electric dipole moment, molecular polarizability, anisotropy of polarizability and molecular first hyperpolarizabiliy of present compound were investigated. The α and β values of Gaussian output are in atomic units (a.u.) so they have been converted into electronic units (esu) (α; 1 a. u. = 0.1482 × 10−24 esu, β; 1 a.u. = 8.6393 × 10−33 esu) and given in Table.2. The mean polarizability (α), anisotropy of polarizability (∆α) and the average value of the first hyperpolarizabiliy (β) can be calculated using the corresponding Equations. Table 2: Nonlinear optical properties of PS calculated using B3LYP/631+ G(d,p)

Fig.4: Frontier Molecular Orbitals of PS 4.5 MOLECULAR ELECTROSTATIC POTENTIAL Electrostatic potential energy is fundamentally a measure of the strength of the nearby charges, nuclei and electrons, at a particular position. The MEP surface is represented by different colors; blue indicates the strongest attraction and red

αxx

142.6319

βxxx

132.2614

αxy

-1.8378

βxxy

2.2047

αyy

134.7297

βxyy

-112.5705

αxz

-41.4231

βyyy

-200.1725

βxxz

-169.1790

βxyz

-49.2734

αyz αzz

0.5984 113.5284


Page 30

αtot

1.9309

X 10-23 esu

βyyz

∆α

7.6409

X 10-23 esu

βxzz

μx

0.35387

βyzz

μy

-1.03591

βzzz

μz

-0.21633

βtot

μtot

1. 11582 Debye

the NBO analysis. The energy of these interactions can be estimated by the second-order 110.9414 perturbation theory. The intramolecular interactions are observed as increase in electron -22.4117 density (ED) in (C-C) anti- bonding orbital that -37.4319 weakens the respective bonds of aromatic ring (1.97e) clearly demonstrates strong delocalization 2.49882 X 10-30 e.s.u are shown in the Table 3. The occupancy of π bonds is lesser than σ bonds which lead more delocalization. The intermolecular interaction of the σ(C2-C3) and σ(C5-C6) distribute to σ*(C4O11) and σ*(C4-O11) leads to less stabilization of 3.94 and 3.62 kJ/mol respectively. dipole 72.3454

In this work, the calculated value of the moment is found to be 0.73135 Debye. The calculated polarizability and anisotropy of the polarizability of the title molecule is 19.309 x 10-24 esu and 76.409 x 10-24 respectively. The magnitude of the molecular hyperpolarizability β, is one of important key factors in a NLO system. The calculated first hyper polarizability of the title compound is 2.49882 x 10-30esu and which is 3.2 times greater than that of the standard NLO material urea. The theoretical calculation of β component is very useful as this clearly indicates the direction of charge delocalization. Domination of particular component indicates on a substantial delocalization of charges in this direction. Therefore, the largest βxxx value indicates charge delocalization along this direction. The large value of hyperpolarizability (β), which is a measure of the non-linear optical activity of the molecular system, is associated with the intramolecular charge transfer, resulting from the electron cloud movement through π conjugated frame work from electron. The physical properties of these conjugated molecules are governed by the high degree of electronic charge delocalization along the charge transfer axis and by the low band gaps.

The hyperconjugative interactions of π(C1-C2) NBO with the anti-bonding orbitals of π* (C3-C4) and π*(C5-C6) which lead to strong delocalization of 14.79 and 22.40 kJ/mol respectively. The enhanced π*(C3-C4) further conjugates with the anti-bonding π*(C1-C2) and π*(C5-C6) which leads to the enormous stabilization with energy 160.62 and 203.39 kJ/mol.The occupancy of lone pair orbitals of LP(1) O14 and LP(1) O15 are relatively high (1.96 and 1.95 e) when compared to that of the second lone pair orbital of LP(2) O11, LP(2) O14 and LP(2) O15 which is considerably low (1.82, 1.84 and 1.78 eV) respectively, indicating that the latter orbital is extensively involved between lone pair orbital LP(2) O15 with antibonding (C13-O14) π- bond whose charge transfer energy value is 52.23 kJ/mol denotes larger delocalization. Table 3: Second order perturbation theory analysis of Fock matrix in NBO basis for PS. Donor

Orbit

Occu Accept Orbi Occupa E(2) E(i)- F(i,j) panc or t ny E(j) (Kcal (a.u)c y / (a.u)b

4.7 NATURAL BOND ORBITAL (NBO)

mol)a

NBO is a powerful tool to explain the charge transfer or delocalization of charge due to the intra-molecular interaction among bonds and also provides a localized depiction of the electron density over a molecule. The second order fock matrix is carried out to evaluate donor (i) level bonds to acceptor (j) level bond interaction [7] in Sri Sarada International Journal of Multidisciplinary Research Vol : 1, Issue: 1, Jan 2018

Page 31

C1-C2

π

C1-H7

σ

C2-C3

σ

C2-H8

σ

C3-C4

π

C5-C6

σ

C5-C6

π

C5-H9

σ

C6-H10

σ

C13O14

σ σ

O11H12 C19H20 O11 O14 O14 O15 O15 O15 C3-C4

σ LP(2) LP(1) LP(2) LP(2) LP(1) LP(2) π*

1.69 C3-C4 π* 0.4386 14.79 0.27 0.058 960 0 C5-C6 π* 22.40 0.28 0.071 1.98 0.2994 3.79 1.07 0.057 250 C2-C3 σ* 4 1.97 C4172 O11

σ* 0.0203 3.94 5 σ* 3.79 1.97 C1-C6 0.0185 σ* 4.32 912 9 C3-C4 π* 0.0159 22.77 1.61 034 C1-C2 5 π* 13.46 1.97 C5-C6 0.0345 π* 30.33 874 0 C131.70 O14 σ* 0.2946 3.62 152

5 15.50 C4- π* O11 1.97 0.2994 π* 23.74 939 4 C1-C2 σ* 0.3235 3.78 1.98 268 C3-C4 9 σ* 4.23 C1-C6 1.99 0.0185 σ* 3.61 550 9 C3-C4 σ* 0.2946 0.80 1.98 C4-C5 685 5 σ* 5.52 1.97 O110.4386 σ* 5.49 252 H12 0

1.08 0.058 1.08 0.057 1.06 0.061 0.29 0.074 0.29 0.057 0.24 0.078 1.08 0.056 0.29 0.059 0.27 0.074 1.08 0.057 1.05 0.060 1.07 0.056 1.53 0.032 1.29 0.076 0.74 0.057

π* 37.35 0.32 0.105 1.82 C4-C5 0.0159 220 5 O151.96 C16 0.0345 554 0 C3-C4 1.84 370

0.0231 4

1.78 354

0.0523 7

1.95 970

0.0231 4

1.78 354

0.0332 8

1.61 034

0.4386 0

The interaction between the oxygen lone-pair LP(1) O15 and the anti-bonding orbitals σ* C16H17) has been calculated using NBO analysis. It is noted that the energetic contributions (0.76) of hyperconjugative interaction is weak, the E(2) value is chemically significant and can be used as a measure of the intramolecular delocalization. The strengthening and contraction of C-O bond and the elongation of O-H bond are due to rehybridization, which reveals the low value of electron density 0.05237 in the σ*(O11-H12) orbital. This indicates the presence of hydrogen bonding interactions in the title compound.As in the case of the O- bond, σ(O11-H12) conjugation with σ*(C4-C5) leads to less stabilization energy of 5.52 kJ/mol. 4.8 UV-VIS SPECTRA ANALYSIS In order to understand the electronic transition of compound, time- dependent DFT (TD-DFT) calculations on electronic absorption spectrum in solvent was performed by B3LYP/6-31+ G(d,p). In the present study, the maximum absorption wavelength is observed at 295.62 nm with an oscillator strength f = 0.1023 9 (Fig 6). The calculated absorption maxima value is predicted as n→π* transition. The two absorption bands are mainly derived from the contribution of bands π→π* with a maximum (λmax) at 243.04 and 236.12 nm with an oscillating strength f = 0.0013 and f = 0.1660 respectively.

Fig.6: Theoretical UV-Vis spectrum of PS 5. CONCLUSION


Page 32

The optimized structure was calculated using DFT calculations at the level of B3LYP with 6-31+ G (d,p) basis set. geometric parameters of molecule were interpreted and compared with the earlier reported experimental values for a similar compound. The magnetic properties of title molecule are carried out from 1H and 13C NMR spectra in chloroform and calculated by GIAO method. UV–Vis spectral analyses of Propyl salicylate have been analyzed by theoretical calculation. Stability of the molecule arising from hyperconjugative interaction leading to its bioactivity, charge delocalization has been analyzed using NBO analysis. Molecular properties of investigated compounds such as HOMO-LUMO energy gap, dipole moment as well as non linear optical activity have been calculated. It was found that the low HOMO-LUMO energy gap and large hyperpolarizability value provide the evidence for the intramolecular charge transfer interactions of investigated compound. The MEP map shows the negative potential sites are on oxygen atoms as well as the positive potential sites are around the hydrogen atoms.

12) T.F. Ardyukoiva, et al., Atlas of Spectra of Aromatic and Heterocyclic Compounds, Nauka Sib otd., Novosibirsk, 1973. 13) Donald L. Pavia, Gary M.Lampman, George S. Kriz, Introduction to Spectroscopy, ISBN: 0-03-031961-7. 14) B. Balachandran and K. parimala, Spectrochimica Acta part A: Molecular and Biomolecular Spectroscopy, 102(2013) 3051. 15) N.P.G. Roeges, A Guide to the Complete Interpretation of Infrared Spectra of Organic Structures, Wiley, New York, 1994. 16) M. Arivazhagan, S. Jeyavijayan, Spectrochima. Acta A 79 (2011) 376-383. 17) G. Rauhut, P.Pulay, J. Phys. Chem.. 99 (1995) 3093-3100.

REFERENCES 1) James Richard Fromm, “Introduction to Esters”, Organic chemistry and Biochemistry, 1997. 2) O. M. Ameen and . A. Olatunji, “African Journal of Pure and Applied Chemistry”, Vol. 3 7 , pp. 119-125, July 2009. 3) McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc. 4) Li Wan Po.A, Irwin. W.J; “Modelling Decomposition in the Solid State Stability of Salsalate in suspension in the presence of Excipients, 1983, 16(1), 115-23. 5) Goldfrank L.R, ed. Goldfrank’s Toxicologic Emergencies, 8th edition, MC Graw Hill: 2006. 6) www.worldofchemicals.com 7) Y. Zhou, . Matsadiq, . Wu, J. Xiao and J. Cheng, “Propyl 4hydroxybenzoate”, Acta Crystallographica Section E, ISSN 1600-5368, January 2010. 8) Mehmet Cinar, Ali Coruh, Mehmet Karabacak; Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy, 83 (2011) 561– 569. 9) V. Krishnakumar, R.J. Xavier, Indian J. Pure Appl. Phys. 41 (2003) 95–98. 10) K. Furic, V. Mohacek, M. Bonifacic, I. Stefanic, J. Mol. Struct. (1992) 267–270. 11) N.P. Singh, R.A. Yadav, Indian J. Phys. B75 (4) (2001) 347– 352.


Page 33

Synthesis, characterization and computational investigation ofNi(II) Complexof N,N'- Bis(2hydroxybenzylidene)-1,2diaminobenzene Dr.S .Anbuselvi, E.Elavarasi and Dr. V. Jayamani Department of Chemistry, SriSarada College for Women, Salem-16. [email protected]

ABSRTACT: Ni(II) Schiff base complex of N,N′Bis(2-hydroxybenzylidene)-1,2diaminobenzene(HBDB)was synthesized. The synthesized Nickel complexwas subjected to elemental analysis, Infrared, electronic spectral studies, molar conductivity and magnetic moment measurements. The synthesized complex showed an octahedral geometry. Quantum chemical calculations involving geometrical optimization, HOMO,LUMOand NLO properties were also performed using semi empirical methods. Keywords: Synthesis, (HBDB),Ni(II) complex, semiempirical method. 1. INTRODUCTION A large number of Schiff bases and their complexes have been studied for their interesting and important properties, e.g., their ability to reversibly bind oxygen1, catalytic activity in hydrogenation of olefins2 and transfer of an amino group3,photochromic properties4,and complexing ability towards some toxic metals5. The high affinity for the chelation of the Schiff bases towards the transition metal ions is utilized in preparing their solid complexes.Nickel complexes of Schiff base ligands possess a variety of applications in the biological, analytical, clinical, and industrial areas6.In recent times Nickel complexes of Schiff base ligands have gained considerable attention, not only due to their

spectroscopic properties and application7,but also due to their remarkable antifungal,antibacterial and antitumor activities8.Schiff base metal complexes derived from aromatic amines and aromatic aldehydes have a wide variety of applications in many fields, e.g., biological, inorganic and analytical chemistry. Thus, looking to the importance of Schiff bases with essential metals in the form of coordination compounds, Wereport the synthesis, characterization and computational investigation of Ni(II)complex of HBDB. 2. EXPERIMENTAL 2.1. Materials and methods All chemicals employed in the present study were of analytical grade and were used without purification. The purity of compounds was checked routinely by thin layer chromatography. Elemental analyses are performed with a Perkin Elmer 2400 CHN analyzer. Magnetic moment of the Nickel (II) complex was measured at room temperature (28 20C) on a simple Guoy-type balance.Fourier transform infrared(FT-IR)spectra was recorded on Shimadzu FT-IR spectrophotometer using KBr technique in 4000400cm-1 range. Electronic spectra of the Ni(II) complex was recorded usingCyber lab UV-Visible spectrophotometer.Molar conductance of the complex was made on freshly prepared 10-3 M solutions in DMF/waterat room temperature usingElicoconductivitymeasurements 185 conductivity bridge. Semi empirical calculations were carried out using the Guassian 09 program package. 2.2 Synthesis of the Schiff base Ligand (HBDB) The ligand was prepared in similar manner according to the earlier reported procedure (scheme-1)9

Scheme-I


Page 34

2.3 synthesis of metal complex of Ni(II) with ligand(HBDB) HBDB (0.01mol) is mixed with the alcoholic solution of Nickel chloride(0.01mol). The mixture was shaken well, cooled and then refluxed for about 4hours.Pink coloured compound obtained was filtered and dried. The compound was recrystallisedusing the mixture of dichloromethane and rectified spirit (2:1) and dried. Yield=3.8g Melting point> 3000C

Scheme-II Structure of the complex is established by colour, physical constant, conductivity measurement, IR and UV spectral data. 3. Results and Discussion In the present work ligand is synthesized from salicyaldehyde andO-phenylenediamine. The ligand and complexare purified. Conductance, magnetic moment, IR and UV spectrum of the complex are analyzed in detail. Table 1: physical and analytical data of the ligand(HBDB)and its Ni(II) complex Compound

Form ula weig ht (a.m. u)

Colo ur

Conductance measurement The molar conductance value of the complex in the mixture of dichloromethane and rectified spirit lies in the range 0.39×10-3 mho cm2 mol-1 which is quite lower than values expected for a 1:1 electrolyte and reveal their non-electrolytic nature. Magnetic moment The magnetic moment value of 3.12 B.M. measured for the complex lies within the range reported for a d8 system, which contains2 unpaired electrons9-12.This result suggests a spinfree distorted octahedral geometry for this complex. This value is higher than the spin-only value. IR Spectra The IR Spectra of the complex, when compared with that of the free ligand, shows remarkable differences. Selected vibrational bands of the free ligand and the Ni (II)complex are quite useful for determining the mode of coordination of the ligand with the metal ion.

Elemental composition %C

%H

%N

HBDB C20H16N2 O2

316.1 212

Red oran ge

75.8 5 (75. 98)

5.2 0 (5. 10)

8.6 1 (8. 86)

[Ni(HBDB)( H2O)2] C20H18NiN 2O4

408.0 620

Redd ish pink

58.4 5 (58. 81)

4.3 2 (4. 45)

6.7 4 (6. 86)

%M

-

14.1 4 (14. 20)

Wave number (cm-1) Fig. 1 IR spectrum of Fig 2: IR spectrum of HBDB [Ni(HBDB)(H2O)2]


Page 35

Table-2: Compared Vibrational Frequencies of Ligand and Complex Ligand υcm-1 3385 1316 1615 -

Complex υcm-1 1264 512 434 1601 3138

Assignment υcm-1 PhenolicO-H Stretching =C-O Stretching Ni-O Stretching Ni-N Stretching C=N Stretching Free water molecule

The free ligand exhibits υC=N and υAr-o stretching bands at 1615cm-1 and 1316 cm-1 respectively which are shifted to lower frequencies by 14cm1 (1601cm-1)and 52cm-1(1264 cm-1) upon complexation showing the coordination through nitrogen and phenolic oxygen atom. The ligand shows a band at 3385cm-1due to phenolic OH sretchingbut the spectra of complex, there is no band at 3385cm-1.The disappearance of band corresponding to phenolic OH sretching in the IR spectra of complex may be suggested that the –OH hydrogen is deprotonated during the formation of complex. The presence of broad and intense band at 3138cm-1is an evidence for the presence offree water molecule in the complex.The υNi-O andυNi-1 Nare reported at approximately at 512cm and -1 434cm respectively. The synthesized Ni(II) complex also shows new absorption band at 566cm-1 and 430cm-1,which are not present in the free ligand attributed to the existence of υNi-o and υNi-Nstretching respectively10. Electronic spectra

Fig.4 UV spectrum of [Ni(HBDB)(H2O)2]

n the electronic absorption spectrum of HBDB,the bands within the 230-270 nmrange, are due to the excitation of the π electrons (π→π* transitions) of the aromatic rings. The bandwithin the320370nmrange and at 380 nm areassigned to the π→ π* andn→π*transitionsofthe azomethine group respectively, while the longer wavelength band at 460 nm is due to an intramolecular charge transfer (CT transition) involving the whole molecule13-15,16,17,18 (Fig.3). The electronic spectrum of the Ni(II) complex exhibits two d-d absorption bands at 660 nm (15,151cm−1) and339 nm (29,498 cm-1),while the third d-d band is not observed. These bands are 3A (F)→3T (F) assigned to (ʋ2) and 2g 1g 3A (F)→3T (P) (ʋ )transitions respectively.In fact 2g 1g 3 3A (F)→3T (F) (ʋ ) is missing, the positions and 2g 2g 1 assignments of these bands suggestan octahedralenvironment around the Ni(II)ion19-25. The band observed in the range of 345-375 nm is due to existence of a ligand-to-metal charge transfer transition26-28(Fig.4). Computational studies In this computational study all calculations, which include geometry optimizations, energies, reduced masses, vibrational spectra, HOMO, LUMO and NLO properties were performed. Molecular structure of [Ni(HBDB)(H2O)2] The molecular structure of the complex is shown in the Fig-5

Fig. 3 UV spectrum of HBDB


Page 36

O6-C27 O6-Ni45 N7-C13 N7-C23 N7-Ni45 N8-C11 Fig.5 Optimizedstructure of [Ni(HBDB)(H2O)2]

N8-C21

Various theoretically computed energies, rotational constants and dipole moment for the complex are shown in Table-3

N8-Ni45

TABLE-3

C11-C13

Parameters

PM6 Values

Global minimum energy (a.u) Zero point vibrational energy (Kcal/mol) Total energy (Kcal/mol) Translational energy ( Kcal/mol) Rotational energy (Kcal/mol) Vibrational energy (Kcal/mol) Rotational constants (GHZ)

-517.3476641

Dipole moment (Debye)

C12-C14

O1-Ni45 O2-H3 O2-H10 O2-Ni45 O4-H5 O4-H9 O4-Ni45

1.298 3 1.964 4 1.039 3 1.024 6 1.990 5 0.996 0 0.998 7 2.047

H37-O1Ni45 H3-O2-H10 H3-O2-Ni45 H10-O2Ni45 H5-O4-H9 H5-O4-Ni45 H9-O4-Ni45 C27-O6-

C12-H15 C13-C16

198.22908

C14-C17

213.756 0.889

C14-H18

0.889

C16-H19

C16-C17

C17-H20

211.979

C21-H22

0.24884 0.16287 0.10501 2.4282

C21-H35 C23-H24 C23-H25

Table-4 Optimized geometrical parameters for [Ni (HBDB)(H2O)2] Complex Bond angle in (°) Bond length Bond angle in (°) in(Ǻ) O1-C37

C11-C12

124.64 84 101.17 7 94.945 9 98.771 6 103.61 22 104.35 76 93.937 7 123.84

C27-C30C31 C27-C30H33 C31-C30H33 C28-C31C30 C28-C31H34 C30-C31H34 C21-C35C36 C21-C35-

120.518 2 118.040 2 121.440 4 120.093 5 119.825 7 120.080 7 117.527 7 124.563

C25-C26 C25-C27 C26-C28 C26-H29 C27-C30 C28-C31 C28-H32 C30-C31 C30-H33 C31-H34 H35-H36

4 1.315 3 1.949 8 1.382 4 1.368 9 1.862 7 1.423 4 1.326 9 1.849 8 1.412 7 1.448 0 1.389 7 1.089 2 1.428 7 1.415 5 1.086 6 1.383 4 1.087 4 1.089 7 1.104 0 1.449 3 1.093 6 1.436 1 1.422 3 1.438 3 1.391 9 1.089 2 1.421 1 1.407 6 1.086 4 1.390 7 1.086 6 1.087 9 1.427 1

Ni45 C13-N7-C23 C13-N7Ni45 C23-N7Ni45 C11-N8-C21 C11-N8Ni45 C21-N8Ni45 N8-C11-C12 N8-C11-C13 C12-C11-C13 C11-C12-C14 C11-C12H15 C14-C12H15 N7-C13-C11 N7-C13-C16 C11-C13-C16 C12-C14-C17 C12-C14H18 C17-C14H18 C13-C16-C17 C13-C16H19 C17-C16H19 C14-C17-C16 C14-C17H20 C16-C17H20 N8-C21-H22 N8-C21-C35 H22-C21C35 N7-C23-H24 N7-C23-C25 H24-C23C25 C23-C25-C26 C23-C25-C27 C26-C25-C27

18 122.43 60 111.89 01 125.64 62 121.79 39 111.20 70 126.92 52 126.54 71 113.48 12 119.97 17 120.17 55 119.77 74 120.04 63 114.47 23 127.31 51 118.21 00 120.25 78 120.28 28 119.45 91 120.33 88 118.95 76 120.70 17 121.04 53 118.93 87 120.01 57 121.20 58 123.33 26 115.45 30 119.87 67 122.77 96 117.34 37 117.53 33 125.21 31 117.24 32


C37 C36-C35C38 C35-C36C38 C35-C36H39 C38-C36H39 O1-C37C35 O1-C37C40 C35-C37C40 C36-C38C41 C36-C38H42 C41-C38H42 C37-C40C41 C37-C40H43 C41-C40H43 C38-C41C40 C38-C41H44 C40-C41H44 O1-Ni45O2 O1-Ni45O4 O1-Ni45O6 O1-Ni45N8 O2-Ni45O6 O2-Ni45N7 O2-Ni45N8 O4-Ni45O6 O4-Ni45N7 O4-Ni45N8 O6-Ni45N7 N7-Ni45N8

8 117.896 0 121.926 7 118.592 7 119.480 5 123.974 9 116.605 3 119.418 8 119.793 5 120.726 9 119.479 4 120.484 4 117.476 6 122.036 5 120.472 2 119.156 5 120.371 3 74.4999 98.2366 79.9963 95.3956 74.0202 104.419 6 101.170 6 97.5517 82.3303 86.8221 96.1925 88.3982

Page 37

H35-H37 H36-H38 H36-C39 H37-C40 H38-C41 H38-H42 C40-C41 C40-H43 C41-H44

1.429 9 1.381 6 1.091 5 1.435 5 1.415 6 1.086 0 1.376 6 1.087 1 1.090 7

C25-C26-C28 C25-C26H29 C28-C26H29 O6-C27-C25 O6-C27-C30 C25-C27-C30 C26-C28-C31 C26-C28H32 C31-C28H32

121.91 14 118.77 15 119.31 69 123.60 93 116.25 67 120.12 97 120.10 15 120.13 91 119.75 87

Ni45-O6-C27-C30 C27-O6-Ni45-O1 C27-O6-Ni45-O2 C27-O6-Ni45-O4 C27-O6-Ni45-N7 C23-N7-C13-C11 C23-N7-C13-C16 Ni45-N7-C13-C11 Ni45-N7-C13-C16 C13-N7-C23-H24 C13-N7-C23-C25

Table-5 Dihedral angle in (°) Ni45-O1-H37H35 Ni45-O1-H37-C40 H37-O1-Ni45-O2 H37-O1-Ni45-O4 H37-O1-Ni45-O6 H37-O1-Ni45-N8 H3-O2-Ni45-O1 H3-O2-Ni45-O6 H3-O2-Ni45-N7 H3-O2-Ni45-N8 H10-O2-Ni45-O1 H10-O2-Ni45-O6 H10-O2-Ni45-N7 H10-O2-Ni45-N8 H5-O4-Ni45-O1 H5-O4-Ni45-O6 H5-O4-Ni45-N7 H5-O4-Ni45-N8 H9-O4-Ni45-O1 H9-O4-Ni45-O6 H9-O4-Ni45-N7 H9-O4-Ni45-N8 Ni45-O6-C27-C25

Dihedral angle in (°)

7.1250

H15-C12-C14-C17

-179.6202

173.256 0 111.426 6 76.2543 172.529 7 11.3047 98.8419 15.0476 77.3724 168.596 0 3.2933 87.0877 179.507 7 89.2687 137.735 7 56.8189 38.4565 127.266 2 32.5985 48.3183 143.593 6 127.596 6 -

H15-C12-C14-H18

0.1831

N7-C13-C16-C17

-179.5943

N7-C13-C16-H19 C11-C13-C16-C17

0.9149 -0.2202

C11-C13-C16-H19

-179.711

C12-C14-C17-C16

0.0570

C12-C14-C17-H20

179.8559

H18-C14-C17-C16 H18-C14-C17-H20

-179.748 0.0509

C13-C16-C17-C14 C13-C16-C17-H20 H19-C16-C17-C14

0.0214 -179.7754 179.5032

H19-C16-C17-H20

-0.2936

N8-C21-H35-H36

177.5991

N8-C21-H35-H37 H22-C21-H35-H36

-3.7171 -3.4610

H22-C21-H35-H37

175.2227

N7-C23-C25-C26 N7-C23-C25-C27

-171.3284 9.8812

H24-C23-C25-C26

8.6671

H24-C23-C25-C27

-170.1234

C23-C25-C26-C28

-179.4726

Ni45-N7-C23-H24 Ni45-N7-C23-C25 C13-N7-Ni45-O2 C13-N7-Ni45-O4 C13-N7-Ni45-O6 C13-N7-Ni45-N8 C23-N7-Ni45-O2 C23-N7-Ni45-O4 C23-N7-Ni45-O6 C23-N7-Ni45-N8 C21-N8-C11-C12 C21-N8-C11-C13 Ni45-N8-C11-C12 Ni45-N8-C11-C13 C11-N8-C21-H22 C11-N8-C21-H 35 Ni45-N8-C21-H22 Ni45-N8-C21-H35 C11-N8-Ni45-O1 C11-N8-Ni45-O2 C11-N8-Ni45-O4 C11-N8-Ni45-N7 C21-N8-Ni45-O1 C21-N8-Ni45-O2 C21-N8-Ni45-O4 C21-N8-Ni45-N7 N8-C11-C12-C14 N8-C11-C12-H15 C13-C11-C12-C14

15.5766 165.186 162.491 3 120.909 4 65.4052 17.6430 176.509 9 2.8841 5.3062 175.299 8 -0.4639 179.531 4 177.462 3 -2.5423 94.4460 93.6854 169.493 2 -6.6772 83.6678 88.2007 -8.6207 175.208 9 -7.9376 172.067 0 174.986 3 -5.0091 -0.0141 178.866 7 176.575 6 -4.5437 173.134 7 97.9060 88.8825 6.4773 9.9742 85.2029 88.0086 170.413 8 179.727 9 -0.5776 -0.2770

C23-C25-C26-H29 C27-C25-C26-C28

0.3683 -0.5841

C27-C25-C26-C28

179.2568

C23-C25-C27-O6

0.1278

C23-C25-C27-C30 C26-C25-C27-O6

179.3371 -178.6658

C26-C25-C27-C30 C25-C26-C28-C31 C25-C26-C28-H32

0.5434 0.3412 -179.9581

H29-C26-C28-C31 H29-C26-C28-H32

-179.4988 0.2018

O6-C27-C30-C31

178.9935

O6-C27-C30-H33 C25-C27-C30-C31 C25-C27-C30-H33

-0.6123 -0.2721 -179.878

C26-C28-C31-C30

-0.0409

C26-C28-C31-H34 H32-C28-C31-C30

179.8258 -179.7427

H32-C28-C31-H34 C27-C30-C31-C28 C27-C30-C31-H34

0.1240 0.0109 -179.8554

H33-C30-C31-C28 H33-C30-C31-H34

179.6032 -0.2632

C21-H35-H36-H38

178.2234

C21-H35-H36-C39 H37-H35-H36-H38 H37-H35-H36-H39

-1.7302 -0.5502 179.4962

C21-H35-H37-O1

1.9998

C21-H35-H37-C40 H36-H35-H37-O1

-177.6091 -179.3209

H36-H35-H37-C40

1.0702

H35-H36-H38-C41 H35-H36-H38-H42 C39-H36-H38-C41 C39-H36-H38-H42 O1-H37-C40-C41

-0.1455 179.9629 179.8077 -0.0839 179.4330

O1-H37-C40-H43

-0.0134

H35-H37-C40-C41

-0.9298

H35-H37-C40-H43 H36-H38-C41-C40

179.6238 0.3157


Page 38

C13-C11-C12-H15 N8-C11-C12-N7 N8-C11-C12-C16 C12-C11-C13-N7 C12-C11-C13-C16 C11-C12-C14-C17 C11-C12-C14-H18

179.417 5 -0.2040 179.657 1 179.800 3 0.3472 0.0735 179.876 8

H36-H38-C41-H44

-179.7584

H42-H38-C41-C40 H42-H38-C41-H44

-179.7913 0.1346

H37-C40-C41-H38

0.2263

H37-C40-C41-H44 H43-C40-C41-H38 H43-C40-C41-H44

-179.6987 179.6468 -0.2782

Vibrational Assignments: According to the theoretical calculations the [Ni (HBDB)(H2O)2] Complex has 45 atoms. It has 129 normal modes of vibrations. Out of this there are 44 stretching vibrations and 85 bending vibrations.Thefundamental frequencies calculated(semiempirical) for the [Ni(HBDB)(H2O)2] show good agreement with experimental values. A small difference between experimental and calculated vibrational modes is observed. This discrepancy may be due to the formation of intermolecular hydrogen bonding. Also we note that the experimental results belong to solid phase and theoretical calculations belong to gaseous phase. Molecular

Orbitals

of

Fig.6HOMO

Fig.7 LUMO

[Ni(HBDB)(H2O)2]

The atomic orbital compositions of the FMO for the complex are sketched in Fig .6 and Fig.7. Here the positive phase is red and negative one is green. Anticorrosive and Antimicrobialproperties of molecules are predicted through HOMO-LUMO energy gap. The HOMO, LUMO energies are used to describe the dynamic stability, hardness and softness of a molecule. According to Koopman’s theorem29, the energies of the HOMO and the LUMO orbitals of any molecule are related to the ionization potential, IP, and the electron affinity, EA,by ELUMO = -|EA| EHOMO = -|IP| The hardness of the molecule is given by η=(ELUMO - EHOMO)/2.The softness is the reciprocal of hardness σ = 1/η. Theoretical values of EHOMO,ELUMO, Frontier orbitalenergy gap (∆E),hardness (η), softness (σ) ofthe Ni(II) complexare shown in Table-6 Table.6 Name of the Compound [Ni(HBDB)( H2O)2]

EHOM

ELUM

O

O

(a.u )

(a.u )

0.29 32

0.04 95

Ene rgy gap ∆E( e)

Hardness η(a.u)

Softn ess σ(a.u )-1

6.63 15

0.1219

8.20 68

The lower HOMO-LUMO energy gap implies that the complex possesses high Antimicrobial properties.Prediction of polarizabillity and first order hyperpolarizabilittyThedipolemoment µ, the polarizability α, average polarizability αtot, Anisotropy of polarizability Δα (esu) and the molecular first hyper polarizabilityβ (esu) of the [Ni(HBDB)(H2O)2]are shown in Table-7 Table-7 µx 1.4853 β xxx 4264.7490282 µy µz

1.3188 -1.3967

βxxy βxyy

401.6142322 123 1.166512

µtot αxx

2.4282 481.4415

βyyy βxxz

3286.4720462 -143.7715963

αxy αyy

36.9047 411.2285

β xyz βyyz

34.338558 103.1621723

αxz

4.2921

βxzz

-34.4704229


Page 39

αyz

4.3117

βyzz

- 4.5002911

αzz

76.5091

βzzz

29.8691259

αtot

323.0597

βx

5461.4451

Δα

380.3484

βy

3683.587

βz

-10.7403

β tot

6587.5870

The non-linear optical parameters such as dipole moment, polarizability, anisotropy of polarizability and first order hyper polarizability of the complex are calculated using semi empirical (PM6) method30,31.The polarizability and the hyper polarizability tensors can be obtained by a frequency job output file of Gaussian. However α and β values of Gaussian output are in atomic units (a.u). So they have been converted into electronic units (esu). It is well known that the higher values of dipole moment, molecular polarizability and hyper polarizability are important for more active NLO properties. Urea is one of the prototypical molecules used in the study of the NLO properties of molecular systems. Therefore it is used frequently as a threshold value for comparative purposes It is well known that the higher values dipole moment, molecular polarisability and hyper polarisability are important for more active NLO properties.Thus the dipole moment and first hyper polarizability of the [Ni(HBDB)(H2O)2] is approximately 1.54 and 165.79 times greater than those of urea.(µ and β of urea are 1.5285 debye and 343.272×10-23esu respectively). Therefore the complex possesses more active NLO properties32,33. 4. Conclusion Based on the physical data, conductance measurement, magnetic moment study, spectral data and computational calculations the structure of the [Ni(HBDB)(H2O)2]complexis assigned . While comparing to urea and [Ni(HBDB)(H2O)2] complex, the hyper polarisability of the complex is greater than those of urea as well as the ligand HBDB. Therefore, the complex acts as better NLO material. Also lower HOMO-LUMO energy gapsuggests that the complex might possesses high antimicrobial properties.

References: [1] Jones R.D.; Summerville D.A.; Basolo F,Chem.Rev.79, 139, 1979. [2] Olie G.H.; Olive S.;“The Chemistry of the CatalyzesHydrogenation of Carbon Monoxide”,p.152, spiringer, Berline, 1984. [3] Dugas H.; Penney C,“Bioorganic chemistry”, p.435, Springer ,New York, 1981. [4] Margerum

J.D.;

Mller

L.J,“photochromism”,p.

569,WileyInterscience, New York, 1971. [5] Sawodny W.J.;Riederer M,Angew. Chem. Int. Edn. Engl. 16, 859, 1977. [6] Kumar S.;Dhar D.N.; Saxena P.N,J.Sci. Ind. Res.68,181187, 2009. [7] Spange S.; Vilsmeier E.; Adolph S.; Fahrmann A, J.Phys. Org. Chem. 12, 547-556, 1999. [8] TumerM.;

Koksal,

H.;Serin

S.;

DigrakM,Transit.Met.Chem. 24, 13-17, 1999. [9] MathammalS.;Sudha N.;Anbuselvi S, Int.J. current res. and modern edu.28-30, 2017. [10] Dong W.K, Polyhedran. 29, 2087-2097, 2010. [11] . AwadM.K.; IssaR.M.; AtlamF.M, materials and corrosion,10,60,2009.C.James, A.. [12] MauryaR.C.; Sharma P.;

RoyS, Synth. React.Inorg.

Met-org.chem, 33, 683, 2003. [13] Krishnankutty, K.; Ummathur, M. B.; Sayudevi, P. J. Argent. Chem. Soc. 96 (1-2), 13-21, 2008. [14] Joshi S.; Pawar V.; Uma V. Int. J. Pharm. Bio Sci.2(1), 240-250, 2011. [15] Singha R.; Mittalb S. P.; Malika S.; Singha R. V. Indian J. Chem. Sect. A. 46, 1406-1413, 2007. [16] Çakır S.; Biçerb E. J. Iran. Chem. Soc. 7(2), 394-404, 2010. [17] Kirkan B.; Gup R. Turk J Chem. 32, 9-17, 2008. [18] Kshash A.H. J. Chem.2(1), 1-5, 2011. [19] Sacconi L. Electronic Structure and Streochemistry of Ni(II) in Transition Metal Chemistry: A Series of Advances. Marcel Dekker. New York: 1968. [20] Osowole A. A.; Kolawole G. A.; Fagade O. E. J. Coord. Chem. 61(7), 1046-1055, 2008. [21] Chandra S.; Gupta K. Indian J. Chem. Sect. A.40(7), 775-779,2001. [22] Lever

A.

B.P.

Spectroscopy.

Modern

2nd

Inorganic

Edition.

Electronic

Amsterdam,

The

Netherlands: Elsevier 1984.


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[23] Jadegoud Y.; Ijare O. B.; Somashekar B. S.; NaganaGowdaG. A.; Mruthyunjayaswamy, B. H. M. J. Coord. Chem., 61(4), 508-5272008. [24] Dawood Z. F.; Mohammed T. J.; SharifM. R.Iraqi J. Vet. Sci.23, 135-14, 2009. [25] Lever A. B. P. Electronic spectra of some transition metal complexes. J. Chem. Edu.45(11), 711-712 1968. [26] Dutta

R.

L.

and

Syamal

A,

Elements

of

Magnetochemistry. 2ndEdition. New Delhi: EastWest Press1993. [27] Maurya R. C.; Chourasia J.; Sharma P. Indian J. Chem. Sect. A, 46, 1594-1604,2007. [28] Ilhan S.; Temel

H.; Yilmaz

I.; Sekerci M.

Polyhedron,26, 2795-2802,2007. [29] Sastri V. S.; Perumareddi R. Corrosion, 53, 671,1996. [30] Sarojini K.; Krishnan

H.; KanakamC.C.; MuthuS;.

Spectrochim. Acta. Part A. 108, 159-170,2013. [31] Muthu S.; Rajamani, T.; Karabacak M.; Asiri A. M. Spectrochim. Acta. Part A. 122, 1-14, 2014. [32] Karpagam J.; Sundaraganesan N.; Sebastian S.; Manoharan S.; Kurt M. J. Raman Spectroscopy, 41, 53-622010. [33] Kurt M.; Karabacak M.; Okur S.; Sayin S.; Yilmaz M.; Sundaraganesan, N.Spectrochim. Acta Part A.94,126-133, 2012.


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DIGITAL IMAGE PROCESSING TECHNIQUES IN FARMING: A SURVEY Dr.R. Uma Rani#1, P.Amsini*2 1

Associate Professor, Dept. of Computer Science,Sri Sarada College for Women(Autonomous), Salem, Tamilnadu, India 2 ResearchScholar, Sri Sarada College for Women(Autonomous), Salem, Tamilnadu, India [email protected],[email protected]

Abstract: Agriculture is the backbone of Indian economy. In rural areas people survive based on farming. But Agriculture growth has been reduced by some problems such as growth in people population, low ground water level, less biodiversity in soil, plant diseases and pest infecting the crop yields. A main challenge is to grow agriculture more worthwhile by increasing its profitability through innovations in digital image processing techniques. Agricultural image processing is one of the fastest growing research areas in recent times. Digital image processing algorithms are used to detect leaf and plant diseases. This paper mainly focus on the survey of different kind of techniques used to improve the agricultural farming from the various problems such as pest infection, plant diseases, and weeds in crop yields and how to reduce the farmer’s manual work. Keywords: Digital Image Processing. 1. INTRODUCTION A significant economic value of Agricultural products includes rice, wheat, potato, tomato, onion, mangoes, sugar-cane, beans, cotton, etc. The economic level of agriculture has been reducing by some important reasons such as the plant diseases and weeds. Digital Image Processing Techniques plays an important role in the agricultural farming to identify the plant diseases, fruit quality management and crop management. The plant leaf infected area is taken

as input image and then neural network is applied to identify the disease affect accuracy. Fruit quality is measured by using machine learning. [1] The common techniques used in farming are image segmentation, crop identification, remote sensing, fruit quality measurement and disease classification. These techniques are easy to use in farming practices and reduce the manual monitoring. [2] 2. LITERATURE REVIEW A research was done on several papers, describing a various methods applied to detect pest, plant diseases and weed in crop yields. These various methods are familiarized with the existing methods. 2.1 Agriculture Role in Indian economy Agriculture is the backbone of the Indian economy. Today India, agricultural research is contemplating as one of the largest research works in the world. Agricultural farm is suffering from people population, less groundwater level, natural disasters, pest and plant diseases and weeds in crop yields. The number of cultivators decreases from 127.3 million to 118.7 million (Census 2001 of 2011). The modern technology and innovation system is used to improve the progressive scale of agricultural growth. The greenhouse technology is used to farming grown 25 to 30 percent every year and productivity rate is increased. Then house farming through technologies reduce the labor intensive, time, pest infection with atmosphere temperature is controlled [3]. 2.2 Different advanced techniques to detect plant disease The major problem of agricultural growth level is reduced by pest, bacteria and fungus infection in plants and weed in crops. Those cause diseases in greenhouse crops. The three common pests are such as whiteflies, thrips and aphids. Those causes the diseases and spread bacteria in entire plant.


Page 42

Another major problem is weed detection. It plays an important role in crop yields. Weed is spread by birds, winds and other animals and controlled through landholders and use of pesticides. The farmers are fully focused the weeds in crop yields regularly and some weeds species are more noticeable at certain times of the year. Manually detect the disease in plant leaves and remove the weed in crop yields both are very time consuming. H. Al-Hiary et. al [4], in this paper applied various techniques are applied to detect the plant disease. Neural network and k-means cluster are uses to detect the plant diseases and increase the both quantity and quality of the agricultural products. These two techniques are used color and texture features set. Detected the infected region through k-means algorithm and often using texture features find the type of diseases in plant leaves. These approaches are very time consuming and increase the farmer’s income. The K-means algorithm detects the disease with 93 percent accuracy and svm classifiers detect the rice disease with 92.7 percent accuracy. [5] Prof. Sanjay et. al [6] , discussed in their paper to detected the diseased leaf region. Firstly, to taken the input image of the diseased leaf and applied segmentation technique and then texture statistic is measured. Firstly RGB input image is converted to HSV and then masked the green pixels. After that computed the texture features through the color co-occurrence matrix. For this value are used to find if diseased leaf or normal leaf. 2.3 Various Techniques for detect the weeds in crops Amruta A et. al [7], described a method to discover the weed in crops. For that purpose they used morphological operation. First, acquired the input image, after that masked the green pixels, then applied filters to remove the noise and then wavelet transform. Finally, find out the weed and crops accurately and sprayer the pesticides on the weed.Rekha.S et. al [8], the authors survey most algorithms which are used to detect the pest in plants and weed in crop yields.

Those techniques are such as wavelet transforms PCA (principal component analysis) and classification. This automatic technique are classified the weed in crop yields accurately with 85 to 96 percentages. 2.4 Excess use of pesticides Pest and insects cause diseases in plants and the weeds both are controlled by using of pesticides. If the excess use of pesticides it is very harmful to the human beings. Pesticide affects the soil and groundwater. In agricultural farming avoid the use of chemical pesticides. In early stages, the plant diseases are detected by using automatic techniques. These techniques are used to prevent from the plant diseases. It’s reduce the manual work and labor cost.[9] 3. IMAGE PROCESSING TECHNIQUES DETECT THE PLANT DISEASES

TO

Image processing techniques are used to detect the plant, leaf diseases and weeds from crop yields. The five basic techniques are used for reduce manual work and human monitoring entire land. The techniques are image acquisition, preprocessing, segmentation, feature extraction and detection. 3.1 Image acquisition The infected area is acquired by using a digital camera. The input image is defined a twodimensional function f(x, y), where x and y are two spatial coordinates horizontally and vertically. Amplitude f at any pair of coordinates (x, y) is called intensity value at that point of the image. An image is digitized, after that used image processing operations. 3.2 Image preprocessing To enhance the image contrast level, isolating regions or apply filters to remove noise from the input image. This preprocessing is used to improve the image quality and gain the better result.


Page 43

3.3 Image segmentation Image segmentation is a major part of detection of the object from the image. Segmentation is used to subdivide an image into constituent regions. It is most difficult task in the image processing. Plant and leaf diseased region could be a segment of the image accurately in early stage. Different approaches in image segmentation discontinuity based and similarity based approach.

Type of the images

Read the input image

Increase contrast level

Applied segmentation technique

Weed in crops

Pest in leaf

In discontinuity based approaches are included isolated points, line and edge detection. This kind of approach is partitioned or subdivision of an image based on some abrupt changes in the intensity level of the image. The similarity based approach is trying to group of pixels in an image which are similar in some sense. Those techniques are threshold, region growing, region splitting and merging. The image segmentation techniques are such as k-means algorithm, svm classifier, artificial neural network etc. 3.4 Object Recognition and Extraction Detect and extracting the objects from the image by using key point. From the input image find the best key point and applied the multitude of algorithms, then extract the important features such as color, boundary and size. Finally the object is detected accurately from the image. [10] 4. RESULT AND DISCUSSION In this paper, we are taken the three types of the input images such as weed in crops, pest infected and diseased leaf. As a first step, read the input image, and then applied the preprocessing technique for improve the brightness and remove the noise using Gaussian filter of the image and then convert to binary image, then finally applied k-means cluster to segment the image and classified the infected area accurately. The experimental result is shown in figure1.

Disease infected leaf

Figure 1: various inputs are taken and detect the pest and disease infected area and weeds in crops.

5. CONCLUSION In existing, peoples are monitoring the plant diseases and weeds through the manual, which is not possible to detect the diseases in whole plants. This paper presented a survey on recent advanced digital image processing techniques is applied to detect the weed in crops, plant disease monitoring and fruit quality measured. From these automatic methods are helpful to reduce the excess use of pesticides and then less human monitoring the entire agricultural land area. Most important thing is reduced the labor cost to the farmers. Digital image processing techniques has been used to wide range of agricultural area. More new techniques are available and easy to detect the diseases in plant. Taken the data set from the real problem areas by using cameras otherwise data is available in online. This data set is used for practices and invents the new techniques in image processing.


Page 44

References [1] Janwale Asaram Pandurng Santosh S. Lomte , ,“Digital Image Processing Applications in Agriculture: A Survey “,International Journal of Advanced Research in Computer Science and Software Engineering , Volume 5, Issue 3, March 2015 ISSN: 2277 128X. [2] Anup Vibhute ,S K Bodhe , “Applications of Image Processing in Agriculture: A Survey”,International Journal of Computer Applications (0975 – 8887) Volume 52– No.2, August 2012. [3] Lopamudra Lenka Samantaray,” A Study on the Current Trend of Agricultural Productivity in India and its Future Prospects”, International Journal of Humanities Social Sciences and Education (IJHSSE),Volume 2, Issue 4, April 2015, PP 16-21,ISSN 2349-0373 (Print) & ISSN 23490381 (Online). [4] H. Al-Hiary, S. Bani-Ahmad, M. Reyalat, M. Braik and Z. ALRahamneh, “Fast and Accurate Detection and Classification of Plant Diseases”, International Journal of Computer Applications (0975 – 8887), Volume 17– No.1, March 2011. [5] Janavi P, Reesha D’Souza and Suman Antony Lasrado, “A Survey on Best Suited Image Processing Method in Agriculture to Detect Plant Diseases”, International Journal of Latest Trends in Engineering and Technology, Special Issue SACAIM 2016, pp. 232-235, e-ISSN: 2278-621X. [6] Prof. Sanjay B. Dhaygude, Mr.Nitin P.Kumbhar,”Agricultural plant Leaf Disease Detection Using Image Processing”, International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering,Vol. 2, Issue 1, January 2013, ISSN: 2278 – 8875. [7] Amruta A. Aware, Kavita Joshi, “Crop and Weed Detection Based on Texture and Size Features and Automatic Spraying of Herbicides”, Volume-6, Issue-1, Page No (1-7), IJARCSSE 2016. [8] Rekha.S , A. Bhagyalakshmi “ A Survey on Weed and Pest Detection System”,International Journal of Advance Research in Computer Science and Management Studies ,ISSN: 2321-7782 (Online) ,Special Issue, December 2013. [9] Md. Wasim Aktar, Dwaipayan Sengupta and Ashim Chowdhury,”Impact of pesticides use in agriculture: their benefits and hazards”, Published online 2009 Mar. doi: 10.2478/v10102-009-00017,PMCID: PMC2984095. [10] Akriti Parida, Sonal Kothari, Nandhini Vineeth, “Real Time Pest Detection and Identification Using Image Processing”, International Journal of Advanced Research in Computer Science and Software Engineering , Volume 5, Issue 4, 2015 ISSN: 2277 128X . [11] Rajneet Kaur, Miss. Manjeet Kaur, “A Brief Review on Plant Disease Detection using in Image Processing”, International Journal of Computer Science and Mobile Computing, Vol.6 Issue.2, February- 2017, pg. 101-106, IJCSMC, Vol. 6, Issue. 2, February 2017, pg.101 – 106, ISSN 2320–088X, IMPACT FACTOR: 6.017.

[12] “Digital Image Processing”, Third Edition, by Rafael C. Gonzalez and Richard E. Woods, published by Pearson Education, Inc, Publishing as Prentice Hall, Copyright 2008 v. [13] “Digital Image Processing PIKS Scientific Inside”, Fourth Edition, by William K.Pratt, Copyright 2007 by John Wiley & Sons, Inc. [14] “Digital Image Processing” by S. Jayaraman, S. Esakkirajan, T. Veerakumar, Published by McGraw Hill Education (India) Private Limited. Copyright 2009.


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On Characterizations of Fuzzy Rough Bi-Maximal Spaces D. Vidhya

Department of Mathematics, Kalasalingam Academy of Research and Education Anand Nagar, Krishnankoil - 626 126 [email protected]

Abstract : In this paper, the concepts of fuzzy rough bi-closure sets, fuzzy rough biresolvable sets and fuzzy rough bi-maximal spaces are introduced. In this connection the characterizations are established. Keywords: fuzzy rough bi-closure sets, fuzzy rough bi-resolvable sets, fuzzy rough bimaximal spaces, fuzzy rough sub-maximal spaces and fuzzy rough bi-resolvable spaces. 2000 AMS Classification: 54A40, 54E55. 1. Introduction The concept of fuzzy set was introduced by Zadeh [11]. Fuzzy sets have applications in many fields such as information [6] and control [7]. The theory of fuzzy topological spaces was introduced and developed by Chang [1] and since then various notions in classical topology has been extended to fuzzy topological spaces. Pawlak [5] introduced the concept of rough set. The concept of rough topological space was introduced by B. P. Mathew and S. J. John [2]. S. Nanda and S. Majumdar [4] introduced the concept of fuzzy rough set. In this paper, the concepts of fuzzy rough bi-closure sets, fuzzy rough bi-resolvable sets and fuzzy rough bimaximal spaces are introduced. In this connection the characterizations are established. Preliminaries 2 Definition 2.1. [5] Let U be a non empty and let B be a complete subalgebra of a Boolean algebra of P(U) of subsets of U. The pair (U, B) is called a rough universe. Let V = (U,B) be a fixed rough universe. Let R be ( ) B if a relation defined as follows: and only if B, . The elements of R are called rough sets and the elements of B are called exact sets. We identify the element ( ) R with the element B and hence an exact set is a rough set inthe sense of the above identi_cation. But a rough set need not be exact;

for example if Uis any nonempty set, then ( )is a rough set which is not exact. Let ( ) and ( )be any two rough ( ∪ sets. Then ∪ ∪ ) , ( ∩ ∩ ∩ ), if and only if ∩ . Note 2.1 [5] We denote a rough set X with lower approximation and upper approximation ( ). by Defintion 2.2 [3] Let U be a non empty and let B be a complete subalgebra of a Boolean algebra of P(U) of subsets of U. Let L be a lattice. Let X be a ) B2 with rough set. Then ( . A ( ) in X is fuzzy rough set characterized by a pair of maps and with the property that ( ) ( ) for all . The collection of all fuzzy rough sets in X is denoted by FRS(X). Note 2.2 [4] In particular L could be the closed interval [0,1]. Defintion 2.3 [9] A fuzzy rough topology on a rough set X is a family T of fuzzy rough sets in X satisfies the following conditions: (i) ̃ , ̃ . (ii) If , then ∩ . (iii) If for all , then ∪ . Then the pair (X,T) is called a fuzzy rough topological space and any fuzzy rough set in T is called a fuzzy rough open set in X. The complement of a fuzzy rough open A is a fuzzy rough closed set. Defintion 2.4 [9] ( ) in X is A fuzzy rough set characterized by a pair of maps and with the property that ( ) ( ) for all . The collection of all fuzzy rough sets in X is denoted by FRS(X). Definition : 2.5 [9] Let A and B be any two fuzzy rough topological groups. Then is defined by ∩ . Definition 2.6 [8] A fuzzy set 𝜆 in a fuzzy topological space (X,T) is called fuzzy dense if there exists no fuzzy closed set in (X,T) such that 𝜆


Page 46

Characterizations of Fuzzy Rough Bi-Maximal Spaces Definition : 3.1 Let (X,T) be a fuzzy rough topological space. Let ( )be a fuzzy rough set. Then Ais said ( )∪ to be a fuzzy rough bi-closure set if ̃ ( ) . The complement of a fuzzy rough bi-closure set is a fuzzy rough bi-interior set. Notation : 3.1 Let (X,T) be a fuzzy rough topological space. ( )∪ (i) ( ) is denoted by FRBC(A). ( )∩ (ii) ( ) is denoted by FRBI(A). Definition : 3.2 Let (X,T) be a fuzzy rough topological space. Then (X,T) is said to be a fuzzy rough bi-maximal space if for every fuzzy rough bi-closure set is a fuzzy rough open set. Remark : 3.1 Let (X,T) be a fuzzy rough topological space. (i) FRBC(A’)=FRBC(A) (ii) FRBI(A’)=FRBI(A). Definition : 3.3 Let A and B be any two fuzzy rough sets. Then is defined by ∩ . Proposition : 3.1 Let (X,T) be a fuzzy rough topological space. Then the following conditions are equivalent: (i) (X,T) is a fuzzy rough bi-maximal space. (ii) FRBC(A)-A is a fuzzy rough closed set for each fuzzy rough set A. (iii) For each fuzzy rough set A, if FRBI(A) ̃ , then A is fuzzy rough closed. Proof: Definition : 3.4 Let (X,T) be a fuzzy rough topological space. Let Abe any fuzzy rough set. Then Ais said to be a said to be a fuzzy ( ) ̃. rough dense set if Definition : 3.5 Let (X,T) be a fuzzy rough topological space. Then (X,T) is said to be a fuzzy rough sub-maximal space if for every fuzzy rough dense set is a fuzzy rough open set. Proposition : 3.2 Let (X,T) be a fuzzy rough topological space. Then the following conditions are equivalent: (i) (X,T) is a fuzzy rough sub-maximal space. (ii) FRBC (A) A is a fuzzy rough closed set for each fuzzy rough set A. (iii) For each fuzzy rough set A, if FRBI(A) ̃ , then Ais fuzzy rough closed.

Proof : The proof is similar to Proposition 3.1. Definition : 3.6. Let (X,T)be a fuzzy rough topological space. Then Ais said to be a fuzzy rough bi-resolvable set if Aand FRint(A ) are fuzzy rough bi-closure sets. Definition : 3.7. Let (X,T) be a fuzzy rough topological space. Then (X,T) is said to be a fuzzy rough bi-resolvable space if for each fuzzy rough bi-resolvable set is fuzzy rough open. Proposition : 3.3 Let (X,T) be a fuzzy rough topological space. Then the following conditions are equivalent: (i) (X,T) is a fuzzy rough bi-maximal space. (ii) FRBC(A) A and FRBC(FRint(A )) FRint(A ) is a fuzzy rough closed set for each fuzzy rough set A. (iii) For each fuzzy rough set A, if FRBI(A) ̃ and FRBI(FRint(A ))) ̃ then Ais fuzzy rough closed. Proof: (ii) (i) Suppose A and FRint( ) are fuzzy rough biclosure sets then A and FRint( ) are fuzzy rough open set when FRBC(A) –A and FRBC(FRint( )) – FRint( )are fuzzy rough closed set. (i) (iii) Let FRBI(A) ̃ and FRBI(FRint( )) ̃ . Then ̃ . Thus is a fuzzy (FRBI(A’)) FRBC(A) rough bi-closure set. Similarly FRint( ) is a fuzzy rough bi-closure set. Therefore A and FRint( ) are fuzzy rough bi-resolvable sets. By (i), Ais fuzzy rough open. Then Ais fuzzy rough closed. (iii) (ii) By Proposition 3.1, FRBC(A) – A isnfuzzybrough closed. Similarly, Since FRBI(FRBC(FRint( )) –FRint( )) (

(

( )) ∪

(

( ))

( )) ∩

(

( )) ∪

(

(

( ))

( )) (

( )) ∩ (

(

))

∪

(

̃.

(

( )∩ ( ( )∩( ∩

( ( )) ∩ ( )∪ ( ( )))) ( ( ) ( ( )) ∩ ( )) ) ̃∩̃


Page 47

(iii)

(i) Let Abe a fuzzy rough bi-resolvable set. Then Aand FRint( ) are fuzzy rough bi-closure sets, FRBC(A ) ̃ and FRBC(FRint( )) ̃ . Thus FRBI(A) ̃ and FRBI(FRint( )) ) ̃ by (iii) A is fuzzy rough open. Remark 3.2 The following implications are true. Fuzzy rough sub-maximal space Fuzzy rough bi-maximal space Fuzzy rough bi-resolvable space The converse of Remark 3.2 need not be true as shown in the following example: Example 3.1 * + be a non-empty set. Let Let B be complete subalgebra of all subsets of U. Let ( ) where * +and * +and ( ) {̃ ̃ } where and ( ) are defined as ( ) and ( ) ; ( ) , ( ) and ( ) and ( ) and ( ) ; ( ) , ( ) and ( ) . Now, Aand Bare fuzzy rough bi-closure sets and fuzzy rough open sets. Hence (X,T)is a fuzzy rough bi-maximal space but not fuzzy rough sub-maximal space.

[3] T. K. Mondal and S. K. Samanta, Intuitionistic Fuzzy Rough Sets and Rough Intuitionistic Fuzzy Sets, The Journal of Fuzzy Mathematics, 9 (2001), 561582. [4] S. Nanda and S. Majumdar, Fuzzy Rough Sets, Fuzzy Sets and Systems, 45(1992), 157-160. [5] Z. Pawlak, Rough sets, Internat. J. Inform. Comput. Sci, 11(1982)5, 341-356. [6] P. Smets, The Degree of Belief in a Fuzzy Event, Information Sciences, 25(1981), 1-19. [7] M. Sugeno, An Introductory Survey of Fuzzy Control, Information Sciences, 36(1985), 59-83. [8] G. Thangaraj and G. Balasubramanian, On Somewhat Fuzzy continuous functions, J. Fuzzy Math., 11(2003)2, 725-736. [9] D. Vidhya, E. Roja and M. K. Uma, Algebraic Fuzzy Roguh Sheaf Group Formed by Pointed Fuzzy Rough Topological Group, Int. J. Math. and Comp. Appl. Research ,4(2014)1, 51-58. [10] D. Vidhya, E. Roja and M. K. Uma, On Fuzzy Rough BG-Boundary Spaces,Annals of Fuzzy Mathematics and Informatics, 10(2015)4,137-140 [11] L. A. Zadeh, Fuzzy Sets, Information and Control, 8(1965), 338-353.

Example 3.2 * + be a non-empty set. Let B be Let complete subalgebra of all subsets of U. Let ( ) where * +and * +and ( ) is defined as {̃ ̃ }where ( ) ( ) ( ) and ; , ( ) ( ) be and ( ) . Let a fuzzy rough set defined as ( ) and ( ) ( ) ( ) ; , and ( ) . Now, A is fuzzy rough bi-resolvable and fuzzy rough open set and B is a bi-closure set but not fuzzy rough open.Hence (X,T)is a fuzzy rough bi-resolvable space but not fuzzy rough bi-maximal space. References

[1] C. L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl., 24(1968),182190. [2] B. P. Mathew and S. J. John, On Rough Topological Spaces, International Journal of Mathematical Archive, 3(2012) N0. 9, 3413-3421.


Page 48

A View On Intuitionistic Pairwise Regular-Lindelof Spaces

A is an object having the form A  X , A , A 1



2

, for all x  X where

A1 and A 2 are subsets of 1 2 1 X satisfying A  A   . The set A is called 2

the set of members of A while A is called the set of nonmembers of A. Every crisp set A on a non-empty set X is obviously an intuitionistic set

T. Ramya, Dr.M.K.Uma and Dr.E.Roja Department of Mathematics, Sri Sarada College for Women, Salem-636016 Tamil Nadu, India [email protected]

having the form A  X , A , A

Abstract : In this paper, we introduce the 

concept of intuitionistic (i, j)-regular cover and 

intuitionistic Pairwise regular cover. The concept of intuitionistic (i, j)- regular kernel, intuitionistic i-kernel, intuitionistic i-cokernel, 

intuitionistic i- Lindelof , intuitionistic (i, j)regular, intuitionistic (i, j)-almost regular, intuitionistic (i, j)-semiregular, intuitionistic (i, j)

nearly Lindelof are introduced and discussed. Keywords intuitionistic

(i,

j)-regular



cover

and



intuitionistic Pairwise regular cover, intuitionistic (i, j)-regular kernel, intuitionistic ikernel, intuitionistic i-cokernel, intuition

istic i-Lindelof , intuitionistic (i, j)-regular, intuitionistic (i, j)-almost regular, intuitionistic (i, j)-semiregular, intuitionistic (i, j)

nearly Lindelof . 2000 AMS Classification: 54A40,54E55. 1 Introduction The concept of an intuitionistic set was introduced by D.Coker in [2]. The intuitionistic set is the discrete form of intuitionistic fuzzy set. The study of bitopological spaces was _rst initiated by J.C.Kelly in 1963 and thereafter a large number of papers have been done to generalize the topological concepts to bitopological setting. Recently the authors introduced and studied the notion of pairwise 

almost lindelof spaces in bitopological spaces. The purpose of this paper is to de_ne and extend the notion of intuitionistic almost regular

Lindelof property in intuitionistic bitopological spaces, which will call intuitionistic almost 

regular-Lindelof spaces and investigate some of their characterizations. 2 Preliminaries Definition 2.1. [1, 2] Let X be a nonempty _xed set. An intuitionistic set (IS for short)

c

.

Definition 2.2. [1, 2] Let X be a nonempty set and the Intuitionistic sets A and B

A  x, A1 , A 2 , B  x, B1 , B 2 .

in the form Then

(i) A  B iff A  B and B (ii) A = B iff A  B and B  A ; 1

1

2

 A2 ;

 B iff A1  A 2  B1  B 2 ; 2 1 (iv) A  X , A , A ; (iii) A

(v)

 Ai  X ,Ai1 ,Ai 2 ;

(vi)

 Ai  X ,Ai1 ,Ai 2 ;

(vii) A‒ B = A  B (viii) `~  x,  , X and

X ~  x, X , .

Definition 2.3. [1, 2] let X and Y be two nonempty sets and f : X → Y a function (i) If B  X , B , B 1

2

is an intuitionistic set in

Y , then the preimage of B under

B  ,is the intuitionistic set in X f 1 B   X , f 1 B1 , f 1 B2  .

f,denoted by defined by (ii) If

f

1

A  X , A1 , A2 is an intuitionistic set in

X, then the image of A under f,denoted by f(A),is the intuitionistic set in Y defined by

f  A  y, f  A1 , f _  A2 

    

f _ A 2  f A 2C

C

,

where

.

Definition 2.4. [2] An intuitionistic topology(IT)on a nonempty set X is a family  of ISs in X satisfying the following axioms: (i)  ~ and X ~   ,

G1  G2   for any G1 ,G2   ; (iii)  Gi   for any arbitrary (ii)

Gi : i  J    .

family

In this case the ordered pair (X, T) is called an intuitionistic topological space (ITS for short) on X and any intuitionistic set in T is known as an intuitionistic open set in X.The complement A of an intuitionistic open set A is called an intuitionistic closed


Page 49

set(ICS for short) in X. 3 Intuitionistic Pairwise Almost Regular Lindelof Spaces Notation 3.1. Let (X, T) be an intuitionistic topological spaces and

A  X , A1 , A2 be an

intuitionistic set in X. (i) Icl(A) denotes intuitionistic closure of A. (ii) Iint(A) denotes intuitionistic interior of A. (iii) Let

A  X , A1 , A2 be an intuitionistic

set in X. x∈ A means x∈ A1 and x ∉ A 2. Definition 3.1. Let (X, T1 , T2) be an intuitionistic topological space and

A  X , A1 , A2 be an intuitionistic set in X. Then the (i) intuitionistic i-cokernel of A for i = 1, 2 is defined and denoted by IcokerTi(A) = ∪{ G  X , G , G 1

2

: G is

intuitionistic i-closed and G ⊆ A}. (ii) intuitionistic i-kernel of A for i = 1; 2 is defined and denoted by IkerTi(A) = ∩{ K  X , K , K 1

2

: K is

intuitionistic i-open and A ⊇ K}. Definition 3.2. Let (X, T1 , T2) be an intuitionistic topological space and

A  X , A1 , A2 be an intuitionistic set in X. Then A is said to be an (i) intuitionistic i-kernel set for i = 1, 2 if Ikeri(A) = A. (ii) intuitionistic i-cokernel set for i = 1, 2 if Icokeri(A) = A. Remark 3.1. Let (X, T1 , T2) be an intuitionistic topological space. Let

A  X , A1 , A2

be any

intuitionistic set in X. Then (i) If A is an intuitionistic i-kernel set for i = 1, 2 then A is intuitionistic i-cokernel set. (ii) If A is an intuitionistic i-cokernel set then is an intuitionistic i-kernel set. (iii)

I keri ( A) = Icokeri A .

(iv)

I keri ( A) = Ikeri A .

A

(v) Ikeri(ϕ∼) = ϕ∼ and Icokeri(ϕ∼) = . (vi) Ikeri(X∼) = X∼ and Icokeri(X∼) = X∼. (vii) Icokeri(cokeri(A)) = Icokeri(A). Definition 3.3. Let (X, T1 , T2) be an intuitionistic topological space and

A  X , A1 , A2

be an intuitionistic set in X. Then A is said to be an (i) intuitionistic (1, 2)-kernel set if Ikeri(A) = A. (ii) intuitionistic i-cokernel set if Icokeri(A) = A. (iii) pairwise intuitionistic kernel set if it is both intuitionistic (1, 2)-kernel set and intuitionistic (2, 1)-kernel set. Note 3.1. Let (X, T1 , T2) be any intuitionistic bitopological space. Then (i) the complement of an intuitionistic (1, 2)kernel set is an intuitionistic (2, 1)cokernel set. (ii) the complement of an intuitionistic (2, 1)kernel set is an intuitionistic (2, 1)cokernel set. (iii) the complement of a pairwise intuitionistic kernel set is an pairwise intuitionistic cokernel set. Definition 3.4. Let (X, T1 , T2) be an intuitionistic bi-topological space. An intuitionistic set F of X is said to be (i) an intuitionistic kernel in X if it is both an intuitionistic 1-kernel and an intuitionistic 2-kernel in X; (ii) an intuitionistic cokernel in X if it is both an intuitionistic 1-cokernel and an intuitionistic 2-cokernel in X. Definition 3.5. An intuitionistic set S of an intuitionistic bi-topological space (X, T1 , T2) is said to be an intuitionistic (i, j)-regular kernel (resp. intuitionistic (i, j)-regular cokernel) if Ii-ker(Ij-coker(S)) = S (resp. Ii-coker(Ijker(S)) = S). S is called an intuitionistic pairwise regular kernel (resp. intuitionistic pairwise regular cokernel) if it is both intuitionistic (1, 2)-regular kernel and intuitionistic (2, 1)-regular kernel (resp. intuitionistic (1, 2)-regular cokernel and intuitionistic (2, 1)-regular cokernel). The complement of an intuitionistic (i, j)-regular kernel (resp. pairwise intuitionistic intuitionistic regular kernel) set is an intuitionistic (i, j)-regular cokernel (resp. pairwise intuitionistic regular cokernel). Definition 3.6. Let (X, T1 , T2) be an intuitionistic bi-topological space. If a family {G: G  X , G , G 1

2

} of

intuitionistic i-kernel sets in X satisfies , ∪{G:

G  X , G1 , G 2  X ~ } then it is called an intuitionistic i-kernel cover of X. Definition 3.7.


Page 50

An intuitionistic bi-topological space (X, T1 , T2) is said to be an intuitionistic i-Lindelof* if every intuitionistic ikernel cover of X has a countable subcover for each i = 1, 2. Definition 3.8. An intuitionistic bi-topological space X is said to be an intuitionistic (i, j)-regular* if for each point x ∈X and for each intuitionistic i-kernel set V of X containing x, there exists an intuitionistic ikernel set U such that x ∈ U and U ⊇Ij-coker(U) ⊇V . X is said to be an intuitionistic pairwise regular if it is both an intuitionistic (1, 2)-regular* and an intuitionistic (2, 1)-regular*. Definition 3.9. An intuitionistic bi-topological space X is said to be an intuitionistic (i, j)-almost regular* if for each point x ∈ X and for each intuitionistic (i, j)-regular kernel set V of X containing x, there exists an intuitionistic (i, j)-regular kernel set U such that x ∈ U and U ⊇ Ij-coker(U) ⊇ V . X is said to be an intuitionistic pairwise almost regular* if it is both an intuitionistic (1, 2)-almost regular* and an intuitionistic (2, 1)-almost regular*. Definition 3.10. An intuitionistic bi-topological space X is said to be an intuitionistic (i, j)-semi regular* if for each point x ∈ X and for each intuitionistic i-kernel set V of X containing x, there exists an intuitionistic ikernel set U such that x ∈ U and U ⊇Ii-ker(Ij-coker(U)) ⊇ V . X is said to be an intuitionistic pairwise semi regular* if it is both intuitionistic (1, 2)-semi regular* and intuitionistic (2, 1)-semi regular*. Definition 3.11. An intuitionistic bi-topological space X is said to be an intuitionistic (i, j)-nearly Lindelof* (reps. an intuitionistic (i, j)-almost Lindelof*) if for every intuitionistic i-kernel cover {Uα : α ∈ Δ } of X, there exists a countable set { αn : n ∈ N } of Δ such that X = ⋃n∈N Ii-ker(Ij-coker(Uαn)) (resp. X = ⋃n∈N Ij-coker(Uαn)). X is called an intuitionistic pairwise nearly Lindelof* (resp. an intuitionistic pairwise almost Lindelof*)if it is both an intuitionistic (1, 2)-nearly Lindelof* and intuitionistic (2 1)nearly Lindelof*( intuitionistic (1, 2)-almost Lindelof* and intuitionistic (2, 1)-almost Lindelof*). Definition 3.12. Let (X, T1 , T2) be any intuitionistic bi-topological space. An intu-

itionistic i-kernel cover {Uα : α ∈ Δ } of X is said to be an intuitionistic (i, j)-regular cover* if for every α ∈ Δ, there exists an intuitionistic (i, j)regular cokernel set Cα of X such that Cα⊇ Uα and X = ⋃n∈N Ij-coker(Uαn)). The collection {Uα : α ∈ Δ } is called an intuitionistic pairwise regular* cover if it is both intuitionistic (1, 2)-regular* cover and intuitionistic (2, 1)-regular* cover. Definition 3.13. An intuitionistic bi-topological space X is said to be an intuitionistic (i, j)-almost regular Lindelof* if for every (i, j)regular*cover {Uα : α ∈ Δ } of X, there exists a countable set { αn : n ∈ N } of Δ such that X = ⋃n∈N Ij-coker(Uαn). X is called an intuitionistic pairwise almost regular Lindelof* if it is both intuitionistic (1, 2)-almost regular Lindelof* and intuitionistic (2, 1)-almost regular Lindelof*. Proposition 3.1. Let X be an intuitionistic bitopological space. The following conditions are equivalent: (i) X is an intuitionistic (i, j)-almost regularLindelof* ; (ii) for every family {Cα : α ∈ Δ } of intuitionistic Ii-cokernel set of X such that for each α ∈ Δ there exists an intuitionistic (i, j)regular kernel set Aα of X with Aα⊆ Cα and ⋂α ∈ Δ Ij-coker(Aα) = ϕ∼ , there exists a countable subfamily { Cαn : n ∈ N } such that ⋂n ∈ N Ij-coker(Cαn) = ϕ∼; (iii) for every family {Cα : α ∈ Δ } of intuitionistic i-cokernel set of X for which every countable subfamily { Cαn : n ∈ N } satisfies ⋂n ∈ N Ij-ker(Aα) ≠ ϕ∼ , the intersection ⋂α ∈ Δ Ij-coker(Aα) ≠ ϕ∼ for each intuitionistic (i, j)-regular kernel set Aα of X with Aα⊆ Cα . Proof: (i) ⟹ (ii) Let {Cα : α ∈ Δ } be a family of intuitionistic icokernel sets of X such that for each α ∈ Δ there exists an intuitionistic (i, j)regular kernel set Aα of X with Aα⊆ Cα and ⋂α∈Δ Ij-coker(Aα) = ϕ∼. It follows that

X ~    I i  co ker(A )    I i  ker(A ) . Since Cα ⊇ Aα = Ij-ker(Ii-coker(Aα)) = Iicoker(Aα), is

I i  co ker(A )  A  C . that

I i  ker(A )  A  C .Therefore

X ~    I i  ker(A )    C . So, the family { C : α ∈ Δ } is an intuitionistic (i, j)-


Page 51

regular cover* of X because for each α ∈ Δ, the intuitionistic (j, i)-regular cokernel set satisfies the conditions

A of X

A ⊇ C and

X ~    I i  ker(A )    C . By (i), there exists a countable subfamily { C n : n ∈ N } such that

X ~   nN I j  co ker(Cn )   nN I j  ker(Cn ) . Therefore ⋂α ∈ Δ Ij-ker( C ) = ϕ∼ . n

Conversely, let {Uα : α ∈ Δ } be an intuitionistic (i, j)-regular cover* of X. By definition (3.11), for each α ∈ Δ, Uα is an intuitionistic i-kernel set in X and there exists an intuitionistic (j, i)-regular cokernel set Cα of X such that Cα⊇ Uα and

X ~    I i  ker( C ) . The family { U : α ∈ Δ } of an intuitionistic i-cokernel sets of X satisfies the conditions, for each α ∈ Δ there exists an intuitionistic (j, i)regular kernel set Cα of X such that

C ⊇ U

and

for which every countable subfamily { N

}

satisfies

the intersection

C n : n ∈

 nN I j  ker( C n )   ~ ,

  I i  co ker( A )   ~

for each intuitionistic (j, i)-regular kernel set Aα of X with Aα⊇ Cα. Proof: (i) ⟹ (ii) It is obvious by the Denfiition (3.11) and Definition (3.12). Since every intuitionistic (j, i)-regular kernel set X is intuitionistic i-kernel. (ii) ⇔ (iii) Let {Cα : α ∈ Δ } be a family of an intuitionistic (i, j)-regular cokernel sets of X such that for each α ∈ Δ there exists an intuitionistic (j, i)-regular kernel set Aα of X with Aα⊆ Cα and    I i  co ker( A )   ~ . It follows that

X ~    I i  co ker(A )    I i  ker(A ) . Since Aα⊇ Cα = Ij-ker(Ii-coker(Aα)) ⊇ Ii-coker(Aα) , Ii

 ker(A )  A  C .

  (C )    I i  ker(C n )    I i  ker(CTherefore )  ~ X ~    I j  ker( A )    (C ) . So, . By (ii), there exists an countable subsets {α n : n ∈ N } of Δ such that

 nN I j  ker(U n )  ~ ,

the family { U : α ∈ Δ } is

X   nN I j  co ker(U  n ) .

an intuitionistic (i, j)-regular* cover of X by the intuitionistic (i, j)-regular kernel sets of X. By (ii), there exists a countable

(ii) ⟺ (iii): The proof is simple.

subfamily { C n : n ∈ N } such that

that

is

 nN I j  co ker(U n )  ~ and

Proposition 3.2. Let (X, T1 , T2) be an intuitionistic bi-topological space. The following conditions are equivalent: (i) X is an intuitionistic (i, j)-almost regularLindelof* ; (ii) for every intuitionistic (i, j)-regular cover {Uα : α ∈ Δ } of X, by intuitionistic (j, i)-regular kernel sets of X, there exists a countable set {αn : n ∈ N } of Δ such that X   nN I j  co ker(U  n ) . (iii) for every family {Uα : α ∈ Δ } of intuitionistic (i, j)-regular cokernel sets of X such that for each α ∈ Δ there exists an intuitionistic (j, i)-regular kernel set A α of X with Aα⊇ Cα and ⋂α ∈ Δ Ij-coker(Aα) = ϕ∼ there exists a countable subfamily { C  n : n ∈ N } such that

 nN I j  ker( C n )   ~ ; (iv) for every family {Cα : α ∈ Δ } of an intuitionistic (i, j)-regular cokernel set of X

X ~   nN I j  co ker(Cn )   nN I j  ker(Cn ) . Therefore

 nN I j  ker( C n )   ~ .

Conversely, let {Uα : α ∈ Δ } be an intuitionistic (i, j)-regular* cover of X by intuitionistic (i, j)-regular kernel sets of X. Then for each α ∈ Δ, there exists an intuitionistic (j, i)-regular cokernel set Cα of X such that Cα⊆ Uα and X ~     I i  ker( C ) . The family { U : α ∈ Δ } of intuitionistic (i, j)regular cokernel sets of X satisfies the conditions, for each α ∈ Δ there exists an intuitionistic (j, i)-regular kernel set of

X

such

that

C

⊇

C

U

  (C )    I i  ker(C )  ~ .

and By

(iii), there exists a countable subset {α n : n ∈ N } of Δ such that

 n I j  ker(U n )  ~ ,


Page 52

that is

 nN I j  co ker(U  n )  ~ . Therefore

X ~   nN I j  co ker(U  n ) . (iii) ⟺ (iv): The proof is obvious. (ii) ⟹ (i) Let {Uα : α ∈ Δ } be an intuitionistic (i, j)-regular cover* of X satisfies the condition of definition (3.11). Since X is an intuitionistic (i, j)-semi regular*, by the definition the topology Ti is generated by the intuitionistic (i, j)-regular kernel set of X, so assume that Uα is an intuitionistic (i, j)-regular kernel set in X for each α ∈ Δ. Hence {Uα : α ∈ Δ } is an intuitionistic (i, j)-regular cover* of X by intuitionistic (i, j)-regular kernel set of X. By (ii), there exists a countable intuitionistic set {αn : n ∈ N } of Δ such that . This shows that X is an intuitionistic (i; j)-almost regular-Lindelof*. Proposition 3.3. Let X be an intuitionistic (i, j)almost regular* space. Then for each x ∈ X and for each intuitionistic (i, j)-regular kernel set W of X containing x, there exists two intuitionistic (i, j)-regular kernel sets

U  X ,U 1 ,U 2 and V  X ,V 1 ,V 2 of X such that x ∈ U and U ⊆ Ij-coker(U) ⊇ V and V Ijcoker(V ) ⊇ (W). Proof: Let x ∈ X and W be an intuitionistic (i, j)-regular kernel set of X containing x. Since X is an intuitionistic (i, j)-almost regular*, there is an intuitionistic (i, j)regular kernel set V such that x ∈ V ⊇ Ij-coker(V ) ⊇ (W). Again, since V is an intuitionistic (i, j)-regular kernel set in X containing x and X is an intuitionistic (i, j)-almost regular*, there exists an intuitionistic (i, j)-regular kernel set U such that x ∈ U ⊇ Ij-coker(U) ⊇ V . So x ∈ U ⊇ Ij-coker(U) ⊇ V ⊇ Ij-coker(V ) ⊇ W. Proposition 3.4. An intuitionistic (i, j)-almost regular Lindelof* and intuitionistic (i, j)-almost regular* space X is an intuitionistic (i, j)-nearly Lindelof*. Proof: Let {Uα : α ∈ Δ } be an intuitionistic (i, j)-regular kernel cover* of X. For each x ∈ X, there exists αx∈ Δ such that x ∈ U αx . Since X is an intuitionistic (i, j)-almost regular*, there exists two intuitionistic (i, j)-regular kernel sets

Vx  X ,Vx ,Vx 1

2

and

Wx  X ,Wx ,Wx 1

2

of X such that x ∈ Vx

⊇ Ij-coker( Vx ) ⊇W ⊇ Ij-coker( W x ) ⊇ ( U  x ) by proposition (3.3). Since for each α ∈ Δ, there exists an intuitionistic (i, j)-regular co kernel set Ij-coker( Vx ) in X such that Ij-coker( Vx ) ⊇ W x and

X ~   xX (Vx )    I j  ker( I i  co ker(Vx )) , the family {Wαx : x ∈ X } is an intuitionistic (i, j)-regular* cover of X by intuitionistic (i, j)-regular kernel sets of X. Since X is an intuitionistic (i, j)-almost regular lindelof*, there exists a countable set of points x1, x2,…., xn…. of X such that

X ~   nN I j  co ker(Wxn ))   nN (Vxn )

by proposition (3.2). Therefore X is an intuitionistic (i, j)-nearly Lindelof*. Proposition 3.5. An intuitionistic (i, j)-semi regular* space X is an intuitionistic (i, j)-nearly Lindelof* if and only if it is an intuitionistic i-Lindelof*. Proof: Let = {Uα : α ∈ Δ } be an intuitionistic i-kernel cover of X. For each x ∈ X, there exists αx∈ Δ such that x ∈ U  x . Since X is an intuitionistic (i, j)-semi regular* and ∈ U  x , there exists an intuitionistic ikernel set

U x such that ∈ Vx ⊇ Ii-ker(Ij-coker ( Vx )) ⊇ ( U  x ).

x

So

X ~   xX (Vx )    I i  ker( I j  co ker(Vx ))

. Now

{I i  ker( I j  co ker(Vx )) : x  X }

forms an intuitionistic (i, j)-regular kernel cover of X. Since X is an intuitionistic (i, j)-nearly Lindelof*, there exists a countable subset of points x1, x2,…., xn…. of X such that

X ~   nN I j  co ker(Wxn ))   nN (Vxn )

. Therefore X is an intuitionistic i-Lindelof*. The Converse is Obvious. Definition 3.14. An intuitionistic bi-topological space X is said to be an intuitionistic (i, j)-nearly regular Lindelof* if for every (i, j)regular cover* {Uα : α ∈ Δ } of X, there exists a countable subset { α n : α ∈ Δ } of Δ such that X ~   nN I i  ker( I j  co ker(U x )) . X is called an intuitionistic pairwise nearly regular Lindelof* if it is both intuitionistic


Page 53

(1, 2)-nearly regular Lindelof* and intuitionistic (2, 1)-nearly regular Lindelof*. Proposition 3.6. An intuitionistic (i, j)-nearly regular lindelof* and intuitionistic (i, j)-almost regular* space X is an intuitionistic (i, j)-nearly regular Lindelof*. Proof: Let = {Uα : α ∈ Δ } be an intuitionistic (i, j)regular* kernel cover of X. For each x ∈ X, there exists αx∈ Δ such that x ∈ U  x . Since X is an intuitionistic (i, j)almost regular*, there exists an intuitionistic (i,

regular*, there exists two intuitionistic (i, j)regular kernel sets and of X such that

Vx  X ,Vx ,Vx 1

Wx  X ,Wx ,Wx 1

2

2

and

of X such that

x ∈ Vx ⊇ Ij-coker( Vx ) ⊇W ⊇ Ij-coker( W x ) ⊇ ( U  x ) by proposition (3.3). Since for each α ∈ Δ, there exists an intuitionistic (i, j)-regular co kernel set Ij-coker( Vx ) in X such that Ij-coker(

Vx

)

⊇

W x

and

j)-regular kernel sets

X ~   xX (Vx )    I j  ker( I i  co ker(Vx ))

αx∈ Δ, there exists an intuitionistic (i, j)-regular

, the family {Wαx : x ∈ X } is an intuitionistic (i, j)regular* cover of X by intuitionistic (i, j)-regular kernel sets of X. Since X is an intuitionistic (i, j)-weakly regular lindelof*, there exists a countable set of points x1, x2,…., xn…. of X such that

Vx of X such that x ∈ Vx and Vx ⊇ Ij-coker( Vx )) and Ijcoker( Vx )) ⊇ ( U  x ). Since for each cokernel set Ij-coker( Vx ) in X such that

Ij-coker(

Vx )) ⊇

(

Vx ) and

X ~   xX (Vx )    I i  ker( I j  co ker(Vx ))

X ~   nN I j  co ker(Wxn ))   nN (U xn )

.

. Therefore X ~  I j  co ker(U xn ) and since

Now { U  x : α ∈ Δ } forms an intuitionistic (i, j)-

X is an intuitionistic (i, j)-semiregular* therefore X is an intuitionistic (i, j)-weakly Lindelof*.

regular cover* of X by intuitionistic (i, j)-regular kernel sets of X . Since X is an intuitionistic (i, j)-nearly regular Lindelof*, there exists a countable set of points x1, x2,…., xn…. of X such that X ~   nN (U xn ) . Therefore X is an intuitionistic (i, j)-nearly regular Lindelof*. Definition 3.15. An intuitionistic bi-topological space X is said to be an intuitionistic (i, j)-weakly regular Lindelof* if for every (i, j)regular cover* {Uα : α ∈ Δ } of X, there exists a countable subset { α n : α ∈ Δ } of Δ such that X ~  I j  co ker(  nN U n ) . X is called an intuitionistic pairwise weakly regular Lindelof* if it is both intuitionistic (1, 2)-weakly regular Lindelof* and intuitionistic (2, 1)-weakly regular Lindelof*.

Proposition 3.8. An intuitionistic bi-topological space X is an intuitionistic (i, j)weakly regular-Lindelof* if and only if for every family {Cα : α ∈ Δ } of intuitionistic intuitionistic cokernel sets of X, for each α ∈ Δ there exists an intuitionistic kernel set Aα of X with Aα ⊆ Cα and   I j  co ker( A )   ~ , there exists a countable subfamily {

U x

:

α

∈

Δ

}

such

that

 nN I j  ker( C n )   ~ . Proof: Let {Cα : α ∈ Δ } be a family of an intuitionistic icokernel sets of X such that for each α ∈ Δ there exists an intuitionistic jkernel set Aα of X with Aα⊆ Cα and

  I i  co ker( A )   ~ . It follows that

Proposition 3.7. An intuitionistic (i, j)-weakly regular Lindelof* and intuitionistic (i, j) regular* space X is an intuitionistic (i, j)weakly Lindelof*. Proof: Let {Uα : α ∈ Δ } be an intuitionistic (i, j)-regular* kernel cover of X. For each x ∈ X, there exists αx∈ Δ such that x ∈ U  x . Since X is an intuitionistic (i, j)-almost

X ~    I i  co ker(A )    I i  ker(A ) . Since Aα⊇ Cα = Ij-ker(Ii-coker(Aα)) ⊇ Ii-coker(Aα) , Ii

 ker(A )  A  C .

Therefore

X ~    I j  ker(A )    (C ) . So, the family { U : α ∈ Δ } is


Page 54

an intuitionistic (i, j)-regular* cover of X. Since X is an intuitionistic (i, j)-weaky regular-Lindelof*, there exists a countable subfamily { C n : n ∈ N } such that

X ~   nN I j  co ker(Cn )   nN I j  ker(Cn ) . Therefore

 nN I j  ker( C n )   ~ .

Conversely, let {Uα : α ∈ Δ } be an intuitionistic (i, j)-regular* cover of X. By definition (3.11), for each α ∈ Δ, Uα is an intuitionistic i-kernel set in X and there exists an intuitionistic (j, i)-regular cokernel set Cα of X such that Cα⊇ Uα and

X ~    I i  ker( C ) .

The family { U : α ∈ Δ } of intuitionistic (i, j)regular cokernel sets of X satisfies the conditions, for each α ∈ Δ there exists an intuitionistic (j, i)-regular kernel set of

X

such

that

C

⊇

C

U

and

  (C )    I i  ker(C )  ~ . By the hypothesis, there exists a countable subset {αn : n ∈ N } of Δ such that

 n I j  ker(U n )  ~ ,

that is

 nN I j  co ker(U  n )  ~ . Therefore

X ~   nN I j  co ker(U  n ) . References [1] D.Coker,A note on intuitionistic sets and intuitionistic points,TURKISH. J. math.20(3)(1996). 343-351. [2] D.Coker,An Introduction to Intuitionistic Topological Spaces,Preliminary Report. Ankara,1995.


Page 55

A VIEW ON FUZZY ρHYPERCONNECTEDNESS IN FUZZY MULTISET TOPOLOGICAL SPACES Dr.B. Amudhambigai, M. Rowthri, V. Madhuri Department of Mathematics, Sri Sarada College for Women, Salem - 16, Tamil Nadu, India. [email protected], [email protected], [email protected] Abstract : In this paper, the concept of fuzzy multiset topological space is introduced. Also the concepts of fuzzy ρconnected M-sets, fuzzy ρ-separated Msets and fuzzy ρ-hyperconnected spaces are studied with examples. Some of their properties and equivalent statements related to fuzzy ρ-connected spaces and fuzzy ρ-hyperconnected spaces are established. Keywords: Fuzzy M-sets, Fuzzy M-topological spaces, Fuzzy ρ-closed M-sets, Fuzzy ρ-connected M-sets, Fuzzy ρ-separated M-sets, Fuzzy ρhyperconnected spaces. 2000 AMS Classification: 54A40, 54E55

Introduction It is well known that the concept of fuzzy sets, firstly defined by Zadeh [14] in 1965. Chronologically in 1968 and 1976, Chang [3] and Lowen [8] redound the concept of fuzzy topological spaces to literature substantively by using this conception. Set is a well-defined collection of distinct objects, that is, the elements of a set are pair wise different. If we relax this restriction an allow repeated occurrences of any object in a set, then the mathematical structure is called as multiset. As a generalization of multiset, Yager [13] introduced fuzzy multisets and suggested possible applications to relational databases. An element of a Fuzzy Multiset can occur more than once with possibly the same or different membership values. K.P.Girish and Sunil Jacob John [6] extended this concept to multiset topological space. The concept of hyperconnectedness in a topological space was

introduced by Steen and Seebach [11] in 1978. Then in 1992, Ajmal.N and Kohli [1] studied some of the characterizations and basic properties of the hyperconnected space. Thereafter, several authors devoted their work to study more properties of hyperconnectedness in a topological space. The fuzzy hyperconnected space was studied by Thangaraj.G [12] and Alkhafaji.M.A [2]. Our aim in this paper is to extend the idea of fuzzy connected and fuzzy hyperconnected in fuzzy M-topological space. Preliminaries In this section some preliminaries and definitions related to this paper are studied. Definition 2.1. [9] A domain X is defined as a set of elements from which M-sets are constructed. The M-set space [X]w is the set of M-sets whose elements are in X such that no element in the M-set occurs more than w times. Definition 2.2. [10] A multiset M drawn from the set X is represented by a function count CM : X → N where N = {0, 1,...}. CM (x) = m implies the number of occurrences of the element x in the mset M. A mset M from the set X = {x1, x2, ..., xn} is defined as M = {m1/x1, m2/x2, ..., mn/xn} where mi is the number of occurrences of the element xi ∈ X Definition 2.3. [13] Let X = {x1, ..., xn} is a universal set. A fuzzy set F of X is characterized by a membership function µA : X [0, 1], where µA(x) is the degree of relevance of x to a concept represented by set symbol A. In particular, a fuzzy multiset A is characterized by a higher order function A : X → [0, 1] → N where N is the set of natural numbers. For each x ∈ X, the membership sequence is defined as the decreasingly ordered sequence of elements in µA(x). It is denoted by { , }; Definition 2.4. [9] Let A F M(X) and x A. Then L(x; A) = max { j; µ j (x) 0 . When we dene an operation between two fuzzy multisets, the length of their membership sequences should be set to equal. So if A and B are FMS at consideration, take L(x; A, B) = max { L(x; A), L(x;B)} . When no ambiguity arises we denote the length of membership by L(x).


Page 56

Definition 2.5. [9] Let A and B be two fuzzy multisets. The following are basic relations and operations for fuzzy multisets :

Inclusion : A ⊆ B ⇔µAj(x) ≤ µB j(x), j = 1, ..., L(x), ∀x X. Equality : A = B ⇔ µA j (x) = µjB(x), j = 1, ..., L(x), ∀x X. Addition : A ⊕ B is defined by the addition operation in X × [0,1] for crisp multisets. Namely if A = {(xi, µi), ...., (xk, µk)} and B = {(xp, µp), ...., (xr, µr)} are two fuzzy multisets, then A⊕B = {(xi, µi), ...., (xk, µk), (xp, µp), ...., (xr, µr)}. Union : µ j (x) = µj (x) ∨µj (x), j = 1, ..., L(x), ∀x

X.

Intersection : µ j (x) = µj (x) ∧ µj (x), j = 1, ..., L(x), ∀x

X.

Definition 2.6. [6] Let M and N be two msets drawn from s set X. Then the following are defined: M = N if CM (x) = CN (x) for all x X.

0M , 1 M

τ with C0M (x) = 0, C1M (x) = w;

If µ, ς τ , then µ ∧ ς τ with Cµ∧ς(x) = min{Cµ(x), Cς(x)}, for all x X; If µi τ for i I, then ∨i µi τ with C∨µi (x) = max{Cµ1 (x), Cµ2 (x), ...}, for all x X. If τ is a fuzzy multiset topology on M, then the pair (M, τ ) is a fuzzy multiset topological space (briefly, fuzzy M-topological space). The members of τ are called fuzzy open M-sets. The complement of fuzzy open M-sets are called fuzzy closed M-sets. Example 3.1. Let X = {a, b}, w = 3 and M = {3/a, 2/b}. Let λ1 , λ2 I M be defined by λ1 = {(0.5, 0.4, 0.2)/a, (0.2, 0.1)/b}, λ2 = {(0.7, 0.6, 0.5)/a, (0.2)/b}. Then τ = {0M , 1M , λ1, λ2} with C1M (x) = 3, C0M (x) = 0, Cλ1 (a) = 3, Cλ1 (b) = 2, Cλ2 (a) = 3 and Cλ2 (b) = 1. Clearly τ is a fuzzy M-topology on M. Therefore (M, τ) is a fuzzy M-topological space. Definition 3.2. Let (M, τ ) be a fuzzy Mtopological space and λ I M be any fuzzy M-set in (M, τ ). Then the fuzzy interior of λ is denoted and defined a Fint(λ) =∨{β set} with

I M : β ≤ λ, β is a fuzzy open M-

M ⊆ N if CM (x) ≤ CN (x) for all x X.

CF int(λ)(x) = max{Cβ(x) : β ≤ λ, β is a fuzzy open Mset}

P = M ∪ N if CP (x) = M ax{CM (x), CN (x)} for all x X.

Example 3.2. As in Example 3.1, (M, τ ) is a fuzzy M-topological space. Let µ IM be

P = M ∩ N if CP (x) = M in{CM (x), CN (x)} for all x X.

defined by µ = {(0.9, 0.8, 0.2)/a, (0.3, 0.2, 0.1)/b}. Then Fint(µ) = {(0.7, 0.6, 0.5)/a, (0.2, 0.1)/b}.

P = M ⊕ N if CP (x) = CM (x) + CN (x) for all x X.

CF int(µ)(a) = max{Cβ(a) : β ≤ λ, β is a fuzzy open M-set} = max {3, 3, 3, 3} = 3 and

Fuzzy Multiset Topological Spaces In this section, the concept of fuzzy Mtopological spaces is introduced. Then the concepts of fuzzy ρ-closed M-sets, fuzzy ρconnected M-sets in fuzzy multiset topological spaces are studied with necessary examples. Also, some equivalent statements are studied. Definition 3.1. Let X be a non-empty set of elements. Let [X]w = M be the set of M-sets whose elements are drawn from X such that no element in the M-set occurs more than w times. A fuzzy multiset topology on a set X is a collection τ of fuzzy M-sets in M satisfying the following axioms:

CF int(µ) (b) = max{Cβ (b) : β ≤ λ, β is a fuzzy open M-set} = max {2, 1, 2, 1} = 2. Definition 3.3. Let (M, τ ) be a fuzzy Mtopological space and λ IM be any fuzzy M-set in (M, τ ). Then the fuzzy closure of λ is denoted and defined as Fcl(λ) = {β set} with

I M : λ ≤ β, β is a fuzzy closed M-

CF cl(λ)(x) = min{Cβ(x) : β ≥ λ, β is a fuzzy closed M-set}. Example 3.3. As in Example 3.1, (M, τ ) is a fuzzy M-topological space. Then τ’ =


Page 57

{0M , 1M , 1M − λ1, 1M − λ2, 1M − (λ1 ∨ λ2), 1M − (λ1 ∧ λ2)}. Let µ I M be defined

Fuzzy ∗g-closed M-set if F cl(λ) ≤ µ with CF cl(λ)(x) ≤ Cµ(x) whenever λ ≤ µ

by µ = {(0.3, 0.2, 0.1)/a, (0.7,0.6, 0.4)/b}. Then

with Cλ (x) ≤ Cµ (x) and µ is a fuzzy gˆ-open M-set.

Fcl(µ) = {(0.3, 0.4, 0.5)/a, (0.8, 0.9, 1)/b}.

Fuzzy #g-semiclosed M-set (briefly, F#g-sclosed) if F scl(λ) ≤ µ with CF scl(λ)(x) ≤

CF cl(µ)(a) = min{Cβ(a) : β ≥ λ, β is a fuzzy closed M-set} = min {3, 3, 3, 3} = 3 and CF cl(µ) (b) = min{Cβ (b) : β ≥ λ, β is a fuzzy closed M-set} = min{3, 3, 3, 3}=3. Definition 3.4. A fuzzy M-set λ IM in a fuzzy Mtopological space (M, τ ) is said to be a Fuzzy preclosed M-set if F cl(F int(λ)) λ with CFcl (F int (λ))(x) Cλ(x). Its complement is called fuzzy preopen M-set. Fuzzy semiopen M-set if λ Fcl(Fint(λ)) with Cλ(x) CF cl(F int(λ))(x). Its complement is called fuzzy semiclosed M-set. Fuzzy regular open M-set if λ = F int(F cl(λ)) with Cλ(x) = CF int(F cl(λ))(x). Its complement is called fuzzy regular closed M-set.

Cµ(x) Cλ(x) ≤ Cµ(x) whenever λ ≤ µ and µ is a fuzzy ∗g-open M-set. Fuzzy g˜-closed M-set if F cl(λ) ≤ µ with CF cl(λ) (x) ≤ Cµ (x) whenever λ ≤ µ with Cλ(x) ≤ Cµ(x) and µ is a fuzzy #gs-open M-set. Fuzzy ρ-closed M-set if F pcl(λ) ≤ F int(µ) with CF λ µ with Cλ (x) pcl(λ)(x) ≤ CF int(µ)(x) whenever Cµ (x) and µ is a fuzzy g˜-open M-set. Its complement is called fuzzy ρ-open M-set. Definition 3.8. A fuzzy M-set λ I M in a fuzzy M-topological space (M, τ ) is said to be a fuzzy ρ-clopen (briefly, Fρ-clopen) M-set if it is both fuzzy ρ-closed M-set and fuzzy ρ-open M-set.

Definition 3.5. Let (M, τ) be a fuzzy Mtopological space and λ I M be any fuzzy M-set in (M, τ ). Then the fuzzy pre-closure of λ is denoted and defined as

Example 3.4. Let X = {a, b}, w = 3 and M = {3/a, 2/b}. Let λ I M be defined by λ = {(0.5, 0.4, 0.2)/a, (0.2,0.1)/b} . Then τ = { 0M, 1M, λ} . Clearly τ is a fuzzy M- topology on M . Therefore (M, τ ) is a fuzzy M-topological space. Then fuzzy ρ-closed M-sets are 0M ≤ µ < λ, 1M .

Fpcl(λ) = {β I M : λ ≤ β, β is a fuzzy preclosed M-set}with

Notation 3.1. Let (M, τ ) be a fuzzy Mtopological space. Then

CF pcl(λ)(x) = min{Cβ(x) : β ≥ λ, β is a fuzzy preclosed M-set}.

F ρOM (M, τ ) denotes the family of all fuzzy ρopen M-sets.

Definition 3.6. Let (M, τ ) be a fuzzy Mtopological space and λ I M be any fuzzy M-set in (M, τ ). Then the fuzzy semi-closure of λ is denoted and defined as

F ρCM (M, τ ) denotes the family of all fuzzy ρclosed M-sets.

Fscl(λ) = {β M-set} with

I M : λ ≤ β, β is a fuzzy semiclosed

CF scl(λ)(x) = min{Cβ(x) : β ≥ λ, β is a fuzzy semiclosed M-set}

Definition 3.7. A fuzzy M-set λ I M in a fuzzy M-topological space (M, τ ) is said to be a Fuzzy gˆ-closed M-set if F cl(λ) ≤ µ with CF cl(λ) (x) ≤ Cµ (x) whenever λ ≤ µ with Cλ(x) ≤ Cµ(x) and µ is a fuzzy semiopen M-set.

Definition 3.9.Let (M, τ) be a fuzzy Mtopological space and λ IM be any fuzzy M-set in (M, τ ). Then the fuzzy ρ-interior of λ is denoted and defined as Fρint(λ) = ∨{β open M-set} with

I M : β ≤ λ, β is a fuzzy ρ-

CF ρcl(λ)(x) = max{Cβ(x) : β ≤ λ, β is a fuzzy ρ-open M-set} Definition 3.10. Let (M, τ ) be a fuzzy Mtopological space and λ IM be any fuzzy M-set in (M, τ ). Then the fuzzy ρ-closure of λ is denoted and defined as Fρcl(λ) = ∧{β closed M-set} with

I M : λ ≤ β, β is a fuzzy ρ-


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CF ρcl(λ)(x) = min{Cβ(x) : β ≥ λ, β is a fuzzy ρ-closed M-set} Definition 3.11. Let (M, τ ) be a fuzzy Mtopological space. Any two fuzzy M-sets λ, δ IM are said to be fuzzy ρ-separated M-set if λ + F ρcl(δ) ≤ 1M with Cλ+Fρcl(δ)( x) ≤ C1M (x) = w for all x X and δ + F ρcl(λ) ≤ 1M with Cδ+Fρcl(λ)(x) ≤ C1M (x) = w for all x

X.

Example 3.5. As in Example 3.4, (M, τ ) be a fuzzy M-topological space. Let µ, δ I M be defined by µ = {(0.3, 0.3, 0.1)/a, (0.2)/b}, δ = {(0.4,0.2)/a, (0.1)/b}. Then Fρcl(δ) = {(0.4, 0.2)/a, (0.2)/b} with CF ρcl(δ)(a) = 2 and CF ρcl(δ)(b) = 1. Therefore µ + F ρcl(δ) ≤ 1M with

3 = Cµ+Fρcl(δ)(a) = C1M (a) = 3 1 = Cµ+Fρcl(δ)(b) < C1M (b) = 2 Also Fρcl(µ) = {(0.3, 0.3, 0.1)/a, (0.2)/b} with CF ρcl(µ)(a) = 3 and CF ρcl(µ)(b) = 1. Therefore δ + F ρcl(µ) ≤ 1M with 3 = Cδ+F ρcl(µ)(a) = C1M (a) = 3 1 = Cδ+Fρcl(µ)(b) < C1M (b) = 2 Proposition 3.1. Let (M, τ ) be a fuzzy Mtopological space. Let λ, δ I M be any two fuzzy M-sets in (M, τ ). Then the following statements hold: If λ, δ are fuzzy ρ-separated M-sets and λ1 ≤ λ with Cλ1 (x) ≤ Cλ(x) for all x X and δ1 δ with Cδ1(x) Cδ(x) for all x X. Then λ1 and δ1 are also fuzzy ρ-separated M-sets. If λ, δ are F ρ-clopen M-sets such that λ q˜δ, then λ and δ are fuzzy ρ-separated M-sets. If λ, δ are F ρ-clopen M-sets and if α, β I M are such that α = λ ∧ (1M − δ) with Cα(x) = Cλ∧(1M −δ)(x) for all x X and β = δ ∧ (1M − λ) with Cβ(x) = Cδ∧(1M −λ)(x) for all x X, then α, β are fuzzy ρseparated M-sets. Proof. (i) Since λ1 ≤ λ, F ρcl(λ1) ≤ Fρcl(λ) with CFρcl(λ1)(x) ≤ CFρcl(λ)(x) for all x X. Since λ, δ fuzzy ρ-separated M-sets, λ+F ρcl(δ) ≤ 1M with Cλ+F

ρcl(δ)(x)

≤ w for all x X which implies that λ1 + F ρcl(δ1) ≤ 1M with Cλ1+Fρcl(δ1)(x) ≤ C1M (x) = w for all x X. Similarly δ1 + F ρcl(λ1) ≤ 1M with Cδ1+Fρcl(λ1)(x) ≤ C1M (x) = w for all x X. Hence λ1 and δ1 are also fuzzy ρ-separated Msets. Since λ, δ are fuzzy ρ-closed M-sets, λ = Fρcl(λ) with Cλ(x) = CFρcl(λ)(x) and δ = Fρcl(δ) with Cδ(x) = CFρcl(δ)(x) for all x X. Given that λ q˜δ (i.e) λ + δ ≤ 1M with Cλ+δ(x) ≤ C1M (x) = w for all x X. Then Fρcl(λ) + δ ≤ 1M with Cδ+Fρcl(λ)(x) ≤ C1M (x) = w for all x X and λ + Fρcl(δ) ≤ 1M with Cλ+Fρcl(δ)(x) C1M (x) = w for all x X. Therefore λ, δ are fuzzy ρ-seperated M-sets. If λ, δ are fuzzy ρ-open M-sets, then 1M − λ and 1M − δ are fuzzy ρ-closed M- sets. Since α = λ ∧ (1M − δ), it is clear that α ≤ λ with Cα(x) ≤ Cλ(x) for all x X and α ≤ 1M − δ with Cα(x) ≤ C(1M −δ)(x) for all x X. Then Fρcl(α) ≤ Fρcl(1M − δ) = 1M − δ with CFρcl(α)(x) ≤ C(1M −δ)(x) for all x X. Also Fρcl(α) + δ ≤ 1M with CFρcl(α)+δ(x) ≤ C1M (x) = w for all x X. Since β ≤ δ, β + Fρcl(α) ≤ 1M with CFρcl(α)+β(x) ≤ C1M (x) = w for all x X. Similarly, α + F ρcl(β) 1M with CF ρcl(β)+α(x) C1M (x) = w for all x X. Therefore α, β are fuzzy ρ-separated M-sets. Proposition 3.2. Let (M, τ ) be a fuzzy Mtopological space. Then λ, β I M are fuzzy ρseparated M-sets iff there exist fuzzy ρ-open Msets µ, ς I M such that λ ≤ µ with Cλ(x) ≤ Cµ(x) for all x X, β ≤ ς with Cβ(x) ≤ Cς(x) for all x X and λ + ς ≤ 1M with Cλ+ς(x) ≤ C1M (x) = w, β + µ ≤ 1M with Cβ+µ(x) ≤ C1M (x) = w for all x X. Proof. Let λ, β I M be fuzzy ρ-separated M-sets. Let ς = 1M − F ρcl(λ) with Cς(x) = C1M −Fρcl(λ)(x) and µ = 1M − Fρcl(β) with Cµ(x) = C1M −Fρcl(β)(x). So µ, ς are fuzzy ρ-open M-sets such that λ ≤ µ with Cλ(x) ≤ Cµ(x) for all x X, β ≤ ς with Cβ(x) ≤ Cς(x) for all x X and λ + ς ≤ 1M with Cλ+ς(x) ≤ C1M (x) = w, β + µ ≤ 1M with Cβ+µ(x) ≤ C1M (x) = w for all x X. On the otherhand, let µ, ς be fuzzy ρ-open Msets such that λ ≤ µ with Cλ(x) ≤ Cµ(x) for all x X, β ≤ ς with Cβ(x) ≤ Cς(x) and λ + ς ≤ 1M with Cλ+ς(x) ≤ C1M (x) = w, β + µ ≤ 1M with Cβ+µ(x) ≤ C1M (x) = w for all x X. Since 1M − ς and 1M − µ are fuzzy ρ-closed M-sets, then Fρcl(λ) ≤ 1M − ς ≤ 1M − β with CFρcl(λ)(x) ≤ C1M −β (x) for all x X and Fρcl(β) ≤ 1M − µ ≤ 1M − λ with CFρcl(β)(x) ≤ C1M −λ(x) for all x X. Thus Fρcl(λ) + β ≤ 1M with CF ρcl(λ)+β(x) ≤ C1M (x) =


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w and Fρcl(β) + λ ≤ 1M with CFρcl(β)+λ(x) ≤ C1M (x) = w. Definition 3.12. Any fuzzy M-set λ I M in fuzzy M-topological space (M, τ ) is said to be proper if λ 0M with Cλ(x) for all x X and λ 1M with Cλ(x) C1M (x) = w for all x X. Definition 3.13. A fuzzy M-topological space (M, τ ) is said to be fuzzy ρ-connected if it has no proper fuzzy ρ-clopen M-sets (both fuzzy ρ-open M-set and fuzzy ρ-closed M-set).

λ, δ IM be two fuzzy ρ-closed M-sets. Then λ = 1M − δ with Cλ(x) = C1 −δ(x) and δ = 1M − λ with Cδ(x) = C1M −λ(x) for all x X, which implies that 1M = (1M − δ) + (1M − λ) with C1M (x) = w = C(1M −δ)+(1M −λ)(x) for all x X. But 1M − δ and 1M − λ are fuzzy ρ-open M-sets. This gives contradiction to our assumption. (iii)⇒ (iv)

Definition 3.14. A fuzzy M-topological space (M, τ ) is said to be a fuzzy ρ-separated if it has two fuzzy ρ-open Msets λ1, λ2 0M such that λ1 + λ2 = 1M with Cλ1+λ2 (x) = C1M (x) = w for all x X. If (M, τ ) is not fuzzy ρ-separated, then it is said to be fuzzy ρconnected. Proposition 3.3. Let (M, τ) be a fuzzy Mtopological space. Then the following statements are equivalent: (M, τ ) is a fuzzy ρ-connected. Let λ, δ I M be any two fuzzy ρ-open M-sets of (M, τ ). Then 1M λ + δ with C1M (x) = w Cλ+δ(x) for all x

X.

Let λ, δ I M be any two fuzzy ρ-closed M-sets of (M, τ ). Then 1M λ + δ with C1M (x) = w

Cλ+δ(x) for all x

X.

There is no proper fuzzy M-set λ fuzzy ρ-clopen M-set.

I M which is

Proof. (i) ⇒ (ii) Let (M, τ ) be fuzzy ρ-connected. On contrary, suppose 1M = λ + δ with C1M (x) = w = Cλ+δ(x) for all x X. By Definition 3.13, (M, τ ) is fuzzy ρ-separated. This gives a contradiction. Therefore 1M (x) = w Cλ+δ(x) for all x X.

λ + δ with C1M

(ii)⇒ (iii) Assume (ii). Suppose 1M = λ + δ with C1M (x) = w = Cλ+δ(x) for all x X where

Assume (iii). Suppose that there is a proper fuzzy M-set λ I M which is fuzzy ρ-clopen M-set. Then µ = 1M − λ with Cµ(x) = C1M −λ(x) for all x X. This implies that µ is a fuzzy ρ-closed M-set. Therefore λ + δ = λ + 1M − λ = 1M with Cλ+δ(x) = C1M (x) = w for all x X which gives a contradiction to (iii). (iv) By Definition 3.12, (M, τ ) is fuzzy ρ-connected. Proposition 3.4. Let (M, τ ) be a fuzzy Mtopological space. If 1M λ + δ with C1M (x) = w Cλ+δ(x) for all x X where λ, δ I are fuzzy ρclosed M-sets, then 1M = λ + δ with C1M x) = w Cλ+ δ(x) where λ, δ IM for all x X are fuzzy ρseparated M-sets. Proof. Let us assume the contrary. Suppose that λ, δ I M are fuzzy ρ-separated M- sets where 1M = λ + δ which implies 1M − δ = λ with C1M −δ (x) = Cλ(x) for all x X. Since Fρcl(λ) + δ ≤ 1M which implies Fρcl(λ) ≤ 1M − δ with CFρcl(λ)(x) ≤ C1M −δ (x) for all x X. Then Fρcl(λ) ≤ λ with CFρcl(λ)(x) ≤ Cλ(x) for all x X and also λ ≤ Fρcl(λ) with Cλ(x) ≤ CFρcl(λ)(x) for all x X. Therefore λ = F ρcl(λ) with Cλ(x) = CFρcl(λ)(x) for all x X. Hence λ is a fuzzy ρ-closed M-set. Similarly, µ is a fuzzy ρclosed M-set. Therefore 1M = λ + µ with C1M (x) = w = Cλ+µ(x) where λ, µ are fuzzy ρ-closed M-sets which is a contradiction to our hypothesis. Proposition 3.5. Let (M, τ ) be a fuzzy Mtopological space. Let λ, µ I M be any two fuzzy ρ-separated M-sets of (M, τ ). If δ ≤ λ ∨ µ with Cδ(x) ≤ Cλ∨µ(x) for all x X and δ I M is a fuzzy ρ-connected M-set, then δ ≤ λ with Cδ(x) ≤ Cλ(x) or δ ≤ µ with Cδ(x) ≤ Cµ(x) for all x X. Proof. Let λ, µ I M be fuzzy ρ-separated M-sets and δ ≤ λ ∨ µ with Cδ(x) ≤ Cλ∨µ(x) for all x X where δ I M is a fuzzy ρ-connected M-set. Since δ ∧ λ ≤ λ with Cδ∧λ(x) ≤ Cλ(x) for all x X and δ ∧


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µ ≤ µ with Cδ∧µ(x) ≤ Cµ(x) for all x X, by Proposition 3.1(i) δ ∧ λ and δ ∧ µ are fuzzy ρseparated M-sets. (δ ∧ λ) ∨ (δ ∧ µ) = δ ∧ (λ ∨ µ) = δ with C(δ∧λ)∨(δ∧µ)(x) = Cδ(x) for all x X. But δ is a fuzzy ρ-connected M-set. Therefore δ ≤ δ ∧ λ with Cδ(x) ≤ Cδ∧λ(x) or δ ≤ δ ∧ µ with Cδ(x) ≤ Cδ∧µ(x) for all x X. Hence δ ≤ λ with Cδ(x) ≤ Cλ(x) or δ ≤ µ with Cδ(x) ≤ Cµ(x) for all x X. Fuzzy ρ-Hyperconnected Spaces Definition 4.1. Any fuzzy M-set λ I M in a fuzzy M-topological space (M, τ ) is said to be a fuzzy ρdense M-set if Fρcl(λ) = 1M with CFρcl(λ)(x) = C1M (x) = w for all x X. Definition 4.2. A fuzzy M-topological space (M,τ) is said to be a fuzzy ρ-hyperconnected if every fuzzy ρ-open M-set µ 0M with Cµ(x) 0 for all x X is a fuzzy ρ-dense M-set. Example 4.1. As in Example 3.4, fuzzy ρ-closure of every fuzzy ρ-open M-sets is 1M . Therefore every fuzzy ρ open M-set is fuzzy ρ-dense M-set. Hence (M, τ ) is a fuzzy ρ-hyperconnected space. Definition 4.3. Any fuzzy M-set λ I M in fuzzy M-topological space (M, τ ) is said to be a fuzzy ρnowhere dense M-set if Fρint(Fρcl(λ)) = 0M with CFρint(Fρcl(λ))(x) = 0 for all x X. Example 4.2. As in Example 3.4. Let γ IM be defined by γ = {(0.2, 0.2, 0.1)/a, (0.1, 0.1)/b}. Then F ρcl(γ) = {(0.2, 0.2, 0.1)/a, (0.1, 0.1)/b} with CFρcl(γ)(a) = 3 and CFρcl(γ)(b) = 2. Therefore Fρint(Fρcl(γ)) = 0M with CFρint(Fρcl(γ))(a) = 0 and CFρint(Fρcl(γ))(b) = 0 Definition 4.4. A fuzzy M-set λ IM in a fuzzy Mtopological space (M, τ ) is said to be a Fuzzy ρ-regular open M-set if λ = Fρint(Fρcl(λ)) with Cλ(x) = CFρint(Fρcl(λ))(x) for all x X.

Proof. (i) ⇒ (ii) Let (M, τ ) be a fuzzy ρ-hyperconnected and λ I M . Suppose that λ is not a fuzzy ρ-nowhere dense M-set. Then Fρint(Fρcl(λ)) 0M with CFρint(Fρcl(λ))(x) 0 for all x X. Since (M, τ ) is fuzzy ρ-hyperconnected and F ρint (F ρcl(λ)) is a fuzzy ρ-open M-set, Fρcl (Fρint (Fρcl(λ))) = 1M with CFρcl (Fρint (Fρcl (λ)))(x) = C1M (x) = w for all x X. Since F ρint (Fρcl(λ))) ≤ Fρcl(λ) F ρcl (Fρint (Fρcl(λ))) ≤ Fρcl(Fρcl(λ)) 1M ≤ Fρcl(λ) with C1M (x) = w ≤ CFρcl(λ)(x). Also Fρcl(λ) ≤ 1M with CFρcl(λ)(x) ≤ C1M (x) = w. Therefore Fρcl(λ) = 1M with CFρcl(λ)(x) = C1M (x) = w for all x X. Hence λ is a fuzzy ρ-dense M-set. ⇒ (iii) Let α, β I M be any two fuzzy ρ-open M-sets with α, β 0M . Suppose α+β ≤ 1Mwith Cα+β(x) ≤ C1M (x) = w for all x X. This implies that F ρcl(α) + β ≤ 1M with CF ρcl(α)+β(x) ≤ C1M (x) = w for all x X. Thus α is not fuzzy ρ-dense M-set. Also α ≤ (Fρcl(α))

Fρcl(α)

Fρint(α) ≤ Fρint

with CFρint(α)(x) ≤ CFρint (Fρcl (α))(x) for all x X. Since α is fuzzy ρ-open M-set, 0M α ≤ Fρint (Fρcl(α)) with Cα (x) ≤ CFρint(Fρcl(α))(x) for all x X. Therefore α is not a fuzzy ρ-nowhere dense Mset which is a contradiction to (ii). Thus α + β ≥ 1M with Cα+β(x) ≥ C1M (x) = w. ⇒ (i)

Every fuzzy M-set λ I M is either fuzzy ρ-dense M-set or fuzzy ρ-nowhere dense M-set;

Given that α, β IM be any two fuzzy ρ-open Msets with α, β 0M and α + β ≥ 1M with Cα+β(x) ≥ C1M (x) = wfor all x X. Suppose that (M, τ ) is not fuzzy ρ-hyperconnected space. Then there exists a fuzzy ρ-open M-set µ I M such that µ is not fuzzy ρ-dense M-set (i.e.,) Fρcl(µ) 1M with CFρcl(µ)(x) C1M (x) = w for all x X. Thus 1M − Fρcl(µ) 0M with C1M −F ρcl(µ)(x) 0 for all x X. Therefore 1M −Fρcl(µ) and µ are fuzzy ρopen M-sets such that {1M−Fρcl(µ)} + µ 1M with C{1M − Fρcl(µ)}+µ(x) C1M (x) = w

If α, β I M are two fuzzy ρ-open M-sets, then α + β ≥ 1M with Cα+β(x) ≥

which is a contradiction to our assumption. Hence (M, τ ) is a fuzzy ρ-hyperconnected space.

Fuzzy ρ-semiopen M-set if λ ≤ Fρcl(Fρint(λ)) with Cλ(x) ≤ CFρcl(Fρint(λ))(x) for all x X. Proposition 4.1. Let (M, τ ) be a fuzzy Mtopological space. Then the following statements are equivalent: (M, τ ) is a fuzzy ρ-hyperconnected;

C1M (x) = w for all x X. Sri Sarada International Journal of Multidisciplinary Research Vol : 1, Issue: 1, Jan 2018

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Proposition 4.2. Let (M, τ ) be a fuzzy Mtopological space. Then the following statements are equivalent: (M, τ ) is a fuzzy ρ-hyperconnected; 1M and 0M are the only fuzzy ρ-regular open Msets in (M, τ ). Proof. (i) ⇒ (ii) Let (M, τ ) be a fuzzy ρ-hyperconnected space. Suppose that µ I M is a proper fuzzy ρ-regular open M-set. Thus µ = Fρint (Fρcl(µ)) with Cµ(x) = CFρint (Fρcl (µ))(x) for all x X. This implies that 1M − µ = 1M – Fρint (Fρcl(µ)) with C1M −µ(x) = C1M for all x X. Thus 1M − µ −Fρint (Fρcl (µ))(x) = Fρcl(1M −Fρcl(µ)) with C1M −µ(x) = CFρcl (1M – Fρcl (µ)) (x) for all x X. Since 1M − µ = F ρcl(1M − F ρcl(µ)) 1M with CF ρcl(1M −F ρcl(µ))(x) is fuzzy ρ-open M-set which is a contradiction. Therefore 1M and 0M are fuzzy ρ-regular open Msets in (M, τ ). (ii) ⇒ (i) Let 1M and 0M are the only fuzzy ρ-regular open M-sets in (M, τ ). Suppose that (M, τ ) is not a fuzzy ρ-hyperconnected. Then there exist a fuzzy ρ-open M-set µ 0M such that Fρcl(µ) 1M with CFρcl(µ)(x) C1M (x) = w for all x X. Since µ is fuzzy ρ-open M-set, Fρcl(Fρint (µ)) 1M with CFρcl (Fρint (µ))(x) C1M (x) = w for all x X. Also, since 1M and 0M are the only fuzzy ρregular open M-sets in (M, τ ), Fρcl (Fρint (µ)) = 0M with CFρcl (Fρint (µ))(x) = 0 for all x X. Thus Fρcl (µ) = 0M with CFρcl (µ)(x) = 0 for all x X, where µ = 0M which is a contradiction. Hence (M, τ ) is a fuzzy ρ-hyperconnected. Proposition 4.3. Let (M, τ ) be a fuzzy ρhyperconnected space. Any fuzzy M-set λ I M is fuzzy ρ-semiopen M-set if F ρint(λ) ƒ= 0M with CF ρint(λ)(x) ƒ= 0 for all x X. Proof. Let (M, τ ) be a fuzzy ρ-hyperconnected space and λ IM is fuzzy M-set with Fρint (λ) 0M with CFρint 0 for all x X. Therefore Fρcl (Fρint (λ)(x) (λ)) = 1M with CFρcl (Fρint (λ))(x) = C1M (x) = w for all x X. Thus λ Fρcl (Fρint (λ)). Hence λ is a fuzzy ρ-semiopen M-set. Definition 4.5. A fuzzy M-topological space (M,τ) is said to be a fuzzy ρ-extremally disconnected if the fuzzy ρ-closure of every fuzzy ρ-open M-set is a fuzzy ρ-open M-set.

hyperconnected space, then it is fuzzy ρextremally disconnected space. Proof. Let (M, τ ) be a fuzzy ρ-hyperconnected space. Then for each fuzzy ρ-open M- set µ, Fρcl (µ) = 1M with CFρcl (µ)(x) = C1M (x) = w for all x X. Since 1M is a fuzzy ρ-open M-set, Fρcl (µ) is a fuzzy ρ-open M-set. Hence (M, τ ) is a fuzzy ρextremally disconnected. Conclusion In this paper, we have introduced fuzzy Mtopological spaces and studied the concepts of fuzzy ρ-connected M-sets and fuzzy ρhyperconnected spaces with examples. Further work may be done on fuzzy symmetric groups in fuzzy M-topological spaces via fuzzy ρ-connected sets. Acknowledgments The authors would like to thank the referees for their valuable suggestions towards the betterment of the paper.

References Ajmal N., Kohli, Properties of hyperconnectedned spaces, their mappings into Hausdorff spaces and embeddings into hyperonnecte spaces, Acta Math. Hungar. 60 (1992) 41-49. [2] Alkhafaji M.A. and Alkanee S.A., On Fuzzy Semi Extremely Disconnected in Fuzzy Topological Space, IOSR Journal of Mathematics 5 (2013) 42-45. [3] Chang.C.L., Fuzzy Topological Sapecs, Journal of Mathematical Analysis and Applications, 24, 182 190 (1968). [4] El-Atik A.A., Abu Donia H.M., Salama A.S., On bconnectedness and b- disconnectedness, Journal of the Egyptian Mathematical Society, (2013) 21, 63-67. [5] Ekici E., Generalized Hyperconnetedness, ActaMath. Hungar.,133(1-2) (2011), 14-147. [6] Girish K.P., Sunil Jacob John, On Multiset Topologies, Theory and Applications of Mathematics and Comuter Science, 2 (1) (2012) 37-52. [7] Jaume Casasnovas, Francesc Rossello, Scalar and fuzzy cardinalities of crisp and fuzzy multisets, Department of Mathematics and Computer Science,Universityof the Balearic Islands, Spain. [8] Lowen R., Fuzzy topological spacea and fuzzy compactness, J. Math. Anal. Appl. [9] 56 (1976) 621-633. [10] Sadaaki Miyamoto, Fuzzy Multisets and Their Generalizations, C.S. Calude et al. (Eds.): Multiset Processing, LNCS 2235, pp. 225235, 2001. [11] Sadaaki Miyamoto, Fuzzy Sets, Multisets and Rough Approximations, V.-N. Huynh et al. (Eds.): IUKM 2015, LNAI 9376, pp. 1114, 2015. [12] Steen L.A. and Seebach .A., Jr., Counterexamples in topology, Springer-Verlag, New York. [1]

Proposition 4.4. Let (M, τ ) be a fuzzy Mtopological space. If (M, τ ) is fuzzy ρ-


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Some Properties of Intuitionistic C ∗-Monoid Spaces C.Bavithra Dr.M.K.Uma and Dr.E.Roja

Department of Mathematics, Sri Sarada College for Women, Salem-636016 Tamilnadu, India [email protected] Abstract In this paper the concepts of intuitionistic monoid, intuitionistic monoid structure space, intuitionistic C ∗-monoid space are introduced. Also we introduce the concept of intuitionistic C ∗-monoid continuous functions, the product of a family of intuitionistic C ∗ monoid spaces and some interesting properties are discussed. Keywords: Intuitionistic monoid, Intuitionistic monoid structure space, Intuitionistic C ∗-monoid space, Intuitionistic C ∗-monoid continuous function. 2000 AMS Classification: 54A40,54E55. Introduction The concept of an intuitionistic set was introduced by D.Coker in [2]. The intuitionistic set is the discrete form of intuitionistic fuzzy set. In this paper we introduce the concepts of intuitionistic monoid, intuitionistic monoid structure space, intuitionistic C ∗-monoid space. Also we introduce the concepts of intuitionistic C∗-monoid continuous functions, the product of a family of intuitionistic C ∗ monoid spaces and some interesting properties are discussed. Preliminaries Definition 2.1. [1,2] Let X be a nonempty fixed set.An intuitionistic set (IS for short) A is an object having the form A = x, A1, A2 for all x X, where A1 and A2 are subsets of X satisfying A1 A2 = φ. The set A1 is called the set of members of A, while A2 is called the set of nonmembers of A. Every crisp set A on a non-empty set X is obviously an intuitionistic set having the form (X, A, Ac). Definition 2.2. [1,2] Let X be a nonempty set, A = (x, A1, A2) for all x ∈ X, B = (x, B1, B2) for all x ∈ X be intuitionistic sets on X, and let {Ai : i ∈ J } be an arbitrary family of intuitionistic sets in X, where Ai = (x, A1, A2) for all x ∈ X. C →C (i)A ⊆ B if and only if A1 ⊆ B1 and B2 ⊆ A2

(ii)A = B if and only if A ⊆ B and B ⊆ A (iii)A ⊂ B if and only if A1 ∪ A2 ⊇ B1 ∪ B2 (iv)A = (X, A2, A1) (v) ∪Ai = (x, ∪A1i, ∩A2 i) (vi) ∩Ai = (x, ∩A1, ∪A2) (vii) A − B = A ∩ B (ix) φ∼ = (x, φ, X) and X∼ = (x, X, φ). Definition 2.3. [1,2] Let X and Y be two nonempty sets and f : X → Y a function If B = x, B 1 , B2 for all x X is an intuitionistic set in Y ,then the preimage of B under f,denoted by f −1(B),is an intuitionistic set in X defined by f −1(B) = (x, f−1(B ), f−1(B )). 1 2 If A = x, A1, A2 for all x X is an intuitionistic set in X,then the image of A under f,denoted by f (A),is the intuitionistic set in Y defined by f (A) = (y, f(A1), f−(A2)) where f−(A2) = (f(A2)c)c. Definition 2.4. [1] An intuitionistic topology (IT for short) on a nonempty set X is a family τ of intuitionistic sets in X satisfying the following axioms: φ∼ and X∼ ∈ τ , G1 ∩ G2 ∈ τ for any G1, G2 ∈ τ; ∪Gi ∈ τ for any arbitrary family {Gi | i ∈ J} ⊆ τ. In this case the ordered pair (X, τ ) is called an intuitionistic topological space (IT S for short) and any intuitionistic set in τ is known as an intuitionistic open set (IOS for short) in X. The complement A of an intuitionistic open set A is called an intuitionistic closed set (ICS for short) in X. Definition 2.5. [1] Let (X, τ ) be an intuitionistic topological space. If a family {(x, G1, G2) : i ∈ J } of IOS’s in X satisfies the condition A ⊆ ∪{(x, G1, G2) : i ∈ J }, then it is called an intuitionistic open cover of A. A finite subfamily of an intuitionistic open cover {(x, G1, G2) : i ∈ J} of A, which is also an intuitionistic open cover of A, is called a finite intuitionistic subcover of {(x, G1, G2) : i ∈ J}. An intuitionistic set A = x, A1, A2 in an intuitionistic topological space (X, τ ) is called intuitionistic compact if and only if each intuitionistic open cover of A has a finite intuitionistic subcover. Definition 2.6. [1] Let (X, τ ) be an intuitionistic topological space and A = x, A1, A2 be an intuitionistic set in X.Then the intuitionistic interior and intuitionistic closure of A are defined by cl(A) = K : K is an intuitionistic closed set in X and A K , int(A) = G : G is an intuitionistic open set in X and G A . It can be also shown that cl(A) is an intuitionistic closed set and int(A) is an intuitionistic open


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set in X and A is an intuitionistic closed set in X if cl(A) = A and A is an intuitionistic open set in X if int(A) = A. Properties of intuitionistic C ∗-Monoidspaces Definition 3.1. Let M be a monoid. An intuitionistic set A = x, A1, A2 in M is called an intuitionistic monoid on M if it satisfies the C →C following conditions: a.b ∈ A for all a, b ∈A (a.b).c= a.(b.c) for all a,b,c∈A a.e = e.a = a for identity e ∈ A. Notation 3.1. Let M be a monoid and A = x, A1, A2 be an intuitionistic set of M . A is an intuitionistic monoid means A1 is a monoid and A2 is not a monoid. Definition 3.2. Let M be a monoid. A family M of intuitionistic monoids in M is said to be intuitionistic monoid structure on M if it satisfies the following axioms: φ∼ and X∼ ∈ M, M1 ∩ M2 ∈ M for any M1, M2 ∈ M; ∪Mi ∈ M for any arbitrary family {Mi | i ∈ J } ⊆ M. Then the ordered pair (X, M) is called an intuitionistic monoid structure space. Every member of M is called an intuitionistic open monoid in (M, M). The complement A of an intuitionistic open monoid A in (M, M) is an intuitionistic closed monoid in (M, M). Example 3.1. Let M = 0,1,2 be a set of integers of module 3 with the binary operation as follows: . 0 1 2 0 0 0 0 1 0 1 2 2 0 2 1 Then (M, .) is a monoid. Define intuitionistic monoid A of M by A = (x, {0, 1}, {2}). Clearly the family M = {φ∼, X∼, A} is an intuitionistic monoid structure. Clearly, the ordered pair (M, M) is called an intuitionistic monoid structure space. Definition 3.3. Let (M, M) be any intuitionistic monoid structure space and A be an intuitionistic monoid in M. Then A is said to be an intuitionistic compact monoid in (M, M) if for every family of {Ai | i ∈ J } of intuitionistic open monoids in (M,M) satisfies the condition A ⊆ ∪i∈J Ai, there exists a finite subfamily J0 = {1, ....n} ⊆ J such that A ⊆ ∪n A . i

Definition 3.4. A map C ∗ : defined on the collection of intuitionistic compact monoids is called an intuitionistic C ∗-monoid operator on M and the pair (M, C ∗) is called an intuitionistic C ∗-monoid space if the following axioms are satisfied: (i) C ∗(φ∼) = φ∼

(ii) A ⊆ C ∗A for every A ⊆ C iii A ⊆ B implies C ∗A ⊆ C ∗B for any A, B ⊆ C. Example 3.2. Let M = {a, b, c}. Let P = ({a}, {b, c}), Q = ({c}, {a, b}), R = ({a, b}, {c}), S = ({a, c}, {b}) and define T (M) = {M∼ , φ∼ , P, Q, R, S}. Now C = {φ∼, P, Q, S} is the collection of intuitionistic compact monoid sets in M . A function C ∗ : defined as C ∗(φ∼) = φ∼, C ∗(P ) = M∼, C ∗(Q) = S, C ∗ (S) = X . Thus (M, C ∗) is an ∼ intuitionistic C ∗-monoid space. Definition 3.5. Let (M1, C1∗) and (M2, C2∗) be any two intuitionistic C ∗-monoid spaces. Let f : (M1, C1∗) → (M2, C2∗) be a function. Then f is said to be an intuitionistic C ∗-monoid continuous function if f (C1∗ A) ⊆ C2∗ f (A) for every intuitionistic monoid A of M1. Proposition 3.1. Let (M1, C1∗) and (M2, C2∗) be any two intuitionistic C ∗-monoid spaces. If f : (M1, C1∗) → (M2, C2∗) is intuitionistic C ∗-monoid continuous, then C1∗ f −1 (B) f −1 (C2∗ B) for every intuitionistic monoid B of M2 . Proof: Let B = (x, B 1, B2 ) and B ⊆ M2. Since f is intuitionistic C ∗-monoid continuous, we have f (C1∗ f −1 (B)) ⊆ C2∗ f (f −1 (B)) ⊆ C2∗ B. Therefore, f −1 (f (C1∗ f −1 (B))) ⊆ f −1 (C2∗ B). Hence, C1∗ f −1 (B) ⊆ f−1 (C2∗ B). Proposition 3.2. Let (M1, C1∗), (M2, C2∗) and (M3, C3∗) be intuitionistic C ∗-monoid spaces. If f : (M1, C1∗) → (M2, C2∗) and g : (M2, C2∗) → (M3, C3∗) are intuitionistic C ∗-monoid continuous, then g f : (M1, C1∗) (M3, C3∗) is intuitionistic C ∗-monoid continuous. Proof: Let A be an intuitionistic monoid of M1. Since g ◦ f (C1∗A) = g(f C1∗A) and f is intuitionistic C ∗monoid continuous. g(f(C1∗ A)) ⊆ g(C2∗ f(A)). As g is intuitionistic C ∗-monoid continuous, we get g(C2∗ f (A)) ⊆ C3∗ g(f (A)). Consequently, g ◦ f (C1∗ A) ⊆ C3∗g ◦ f (A). Hence, g ◦ f is intuitionistic C ∗monoid continuous. Proposition 3.3. Let (M1, C1∗) and (M2, C2∗) be intuitionistic C ∗-monoid spaces and let (A, C1∗A ) be an intuitionistic C ∗-closed monoid subspace of (M1 , C1∗ ). If f : (M1 , C1∗ ) → (M2 , C2∗ ) is intuitionistic


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C ∗-monoid continuous, then f

A : (A, C1∗A

)(M2 , C2∗ ) is intuitionistic C ∗-monoid continuous. Proof: If B ⊆ A, then f | A(C1∗A B) = f | A(C1∗ B ∩ A) = f | A(C1∗ B) = f(C1∗ B) ⊆ C2∗ f(B) = C2∗ f | A(B). Hence f | A is intuitionistic C ∗-monoid continuous. Definition 3.6. Let (M1, C1∗) and (M2, C2∗) be any two intuitionistic C ∗-monoid spaces. Let f : (M1, C1∗) → (M2, C2∗) is said to be intuitionistic C ∗closed (open) monoid function if f (F ) is an intuitionistic C ∗-closed (open) monoid of (M2, C2∗) whenever F is an intuitionistic C ∗-closed (open) monoid of (M1, C1∗). Proposition 3.4. A map f : (M1, C1∗) → (M2, C2∗) is intuitionistic C ∗-closed monoid function if and only if for each intuitionistic monoid B of M2 and each intuitionistic C ∗-open monoid G of (M1, C1∗) containing f −1(B), there is an intuitionistic C ∗open monoid U of (M2, C2∗) such that B U and f −1(U ) G. Proof: Suppose that f is intuitionistic C ∗-closed monoid function. Let B be an intuitionistic monoid of M2 and G be an intuitionistic C ∗-open monoid of (M1, C1∗) such that f −1(B) ⊆ G. Then f (G) is an intuitionistic C ∗-closed monoid of (M2,C2∗). XY Let U = f (G ).Then U is an intuitionistic C ∗open monoid of (M2, C2∗) Y −1 −1X XX X (U) = f (f (G ) ) = f −1(f (G )) ⊆ G = G. Therefore, U is an intuitionistic ∗ C -open monoid of (M2, C2∗) containing B such that f(U ) ⊆ G. Conversely, suppose that F is an intuitionistic C ∗closed monoid of (M1,C1∗). Then Y X X f −1(f (F ) ) ⊆ F and F is an intuitionistic C ∗open monoid of (M , C ∗). By hypothesis, there is an intuitionistic C ∗-open monoid U of f

X (M2,C2∗) such that f(F) ⊆ U andf −1(U ) ⊆ F . Therefore, F ⊆ f−1(U) . Consequently, U ⊆ f (F ) ⊆ f(f−1(U) ) ⊆

Y Y U , which implies that f (F) = U . Thus, f (F) is an intuitionistic C ∗-closed monoid of (M2, C2∗). Hence, f is intuitionistic C ∗-closed monoid function. Notation 3.2. The product of a family {(Mα , Cα∗ ) : α ∈ I} of intuitionistic C ∗ monoid spaces, denoted by Πα∈I (Mα , Cα∗ ), is an intuitionistic C ∗ monoid space (Πα∈I Mα , Cα∗ ) where Πα∈IMα denotes the cartesian product of intuitionistic monoids Mα, α I, and C ∗ is the intuitionistic C ∗ monoid operator generated by the projections πα : Πα∈I (Mα , C ∗) → (Mα , C ∗), α ∈ I, that is defined by C ∗ A = Πα∈I Cα∗ πα (A) for each A ⊆ Πα∈IMα. Proposition 3.5. Let {(Mα , Cα∗ ) : α ∈ I} be a family of intuitionistic C ∗ monoid spaces and let β ∈ I. Then F is an intuitionistic C ∗-closed monoid of (Mβ, Cβ) if and only if F Πα∈I Mα is an intuitionistic C ∗-closed monoid of Πα∈I (Mα , Cα∗ ). Proof: Let β I and let F be an intuitionistic C ∗-closed monoid of (Mβ, Cβ). Since πβ is intuitionistic C ∗-monoid continuous, π−1(F ) is intuitionistic C ∗closed monoid of Πα∈I (Mα , Cα∗ ). But π−1(F) = F × Πα∈I Mα , hence F × Πα∈I Mα is intuitionistic C ∗ closed monoid of Πα∈I (Mα , Cα∗ ). Conversely, let F × Πα∈I Mα be an intuitionistic C ∗ - closed monoid of Πα∈I (Mα , Cα∗ ). Since πβ is intuitionistic C ∗-closed monoid, πβ (F × Πα∈IMα) = F is intuitionistic C ∗closed monoid of (Mβ, Cβ). Proposition 3.6. Let {(Mα , Cα∗ ) : α ∈ I} be a family of intuitionistic C ∗ monoid spaces and let β ∈ I. Then G is an intuitionistic C ∗-open monoid of (Mβ, C β) if and only if G Πα∈I Mα is an ∗ intuitionistic C -open monoid of Πα∈I (Mα , Cα∗ ). Proof: Let β I and let G be an intuitionistic C ∗-open monoid of (Mβ, Cβ). Since πβ is intuitionistic C ∗-monoid continuous, π−1(G) is intuitionistic C ∗open monoid of Πα∈I (Mα , Cα∗ ). But π −1 (G) = G × Πα∈I Mα , therefore G × Πα∈I Mα is intuitionistic C ∗open monoid of Πα∈I (Mα , Cα∗ ). Conversely, let G × Πα∈I (Mα , Cα∗ ) be an intuitionistic C ∗-open monoid of Πα∈I (Mα , Cα∗ ). Then Πα∈I Mα − G × Πα∈I Mα is intuitionistic C ∗ - closed monoid of Πα∈I (Mα , Cα∗ ). But Πα∈I Mα − G × Πα∈I Mα = (Xβ − G) × Πα∈I Mα , hence (Xβ − G) × Πα∈I Mα is intuitionistic C ∗-closed monoid of Πα∈I (Mα , Cα∗ ). By


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Proposition (3.5), Xβ − G is an intuitionistic C ∗ closed monoid of (Mβ , Cβ∗ ). Conse- quently, G is an intuitionistic C ∗-open monoid of (Mβ, Cβ∗). Proposition 3.7. Let (M1 , C1∗ ) be an intuitionistic C ∗ monoid space, {(M2α , C2∗α ) : α I be a family of intuitionistic C ∗ monoid spaces and f : (M1,C1∗) Πα∈I(M2α , C2α∗ ) be a map. Then f is an intuitionistic C ∗ closed monoid function if and only if πα f is intuitionistic C ∗ closed monoid for each α I. Proof: Let f be an intuitionistic C ∗ closed monoid function. Since πα is intuitionistic C ∗ closed monoid function for each α I, πα f is also intuitionistic C ∗ closed monoid for each α I. Conversely, let πα f be intuitionistic C ∗ closed monoid function for each α I. Suppose that f is not intuitionistic C ∗ closed monoid function. Then there exists an intuitionistic C ∗ closed monoid F of (M1 , C1∗ ) such that Πα∈I C2∗α πα (f (F )) ¢ f (F ). Therefore, there exists β ∈ I such that C2∗β πβ(f (F )) ¢ π βf (F ). But πβ ◦ f is intuitionistic C ∗ closed monoid of (M2β , C2∗ ). This is a contradiction. Hence f is an intuitionistic C ∗ closed monoid function. Proposition 3.8. Let (M1 , C1∗ ) be an intuitionistic C ∗ monoid space, {(M2α , C2∗α ) : αI be a family of intuitionistic C ∗ monoid spaces and f : (M1,C1∗) Πα∈I(M2α , C2α∗ ) be a map. Then f is an intuitionistic C ∗ monoid continuous function if and only if πα f is intuitionistic C ∗ monoid continuous function for each α I. Proof: Let f be an intuitionistic C ∗ monoid continuous function. Since πα is intuitionistic C ∗ monoid continuous function for each α ∈ I, πα ◦ f is also intuitionistic C ∗ monoid continuous function for each α I. Conversely, let πα f be intuitionistic C ∗ monoid continuous function for each α I. Suppose that f is not intuitionistic C ∗ monoid continuous function. Then there exists an intuitionistic C ∗ monoid A of M1 such that f(C1∗ A) ¢ Πα∈I C2∗α πα (f (A)). Therefore, there exists β ∈ I such that πβ(f (C1∗A)) ¢ C2∗β πβ f (A). This contradicts the continuity of πβ f . Consequently f is an intuitionistic C ∗ monoid continuous function.

References 1. D.Coker, A note on intuitionistic sets and intuitionistic points,Turkish,J.Math.20(3)(1996),343-351. 2. D.Coker, An Introduction to Intuitionistic topological Spaces, Preliminary Re- port,Ankara,1995. 3. J.L.Kelly, General Topology, D.van Nostrand Company, Inc.,1995.


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