An involutive picture fuzzy negator on picture fuzzy sets and some De Morgan triples Bui Cong Cuong
Roan Thi Ngan
Bui Duong Hai
Institute of Mathematics Hanoi,Vietnam Email:
[email protected]
Basic Science Faculty Hanoi University of Natural Resources and Environment Email:
[email protected]
Faculty of Mathematical Economics National Economics University1 Email:
[email protected]
Abstract—A generalization of the Zadeh’s fuzzy sets and Atanassov’s intuitionistic fuzzy sets is the ”Picture fuzzy set”. This concept is effective in approaching problems relating to the information aggregation processes in decision making problems, for examples voting, financial forecasting, estimating risks in business, etc. This promotes vigorously the construction of the picture fuzzy theory: picture fuzzy operators on picture fuzzy sets, picture fuzzy logic. In this paper, we will give some properties of an involutive picture negator and some corresponding De Morgan fuzzy triples on picture fuzzy sets.
I.
I NTRODUCTION
Since Zadeh first described the fuzzy sets (FSs) in 1965 [1], many theories treating imprecision and uncertainty have been introduced. Some of these are extensions of fuzzy set theory [11-15]. In the 1980s, Atanassov proposed a generalization of the fuzzy sets and of the fuzzy logic, intuitionistic fuzzy sets and intuitionistic fuzzy logic [11,14]. Definition 1.1 [11] An intuitionistic fuzzy set (IFS) A on the universe X is an object of the form A = {(x, µA (x), νA (x))|x ∈ X}, where µA (x) ∈ [0, 1] is called the ”degree of membership of x in A”, νA (x) ∈ [0, 1] is called the “degree of non-membership of x in A”, and µA (x) and νA (x) satisfy µA (x) + νA (x) ≤ 1 for all x ∈ X. In this paper, let IF S(X) denote the set of all the intuitionistic fuzzy sets on X. A generalization of fuzzy sets and intuitionistic fuzzy sets are the following notion of picture fuzzy sets. Definition 1.2 A picture fuzzy set A on the universe X is an object of the form A = {(x, µA (x), ηA (x), νA (x))|x ∈ X}, where µA (x) ∈ [0, 1] is called the “degree of positive membership of x in A”, ηA (x) ∈ [0, 1] is called the “degree of neutral membership of x in A”, and νA (x) ∈ [0, 1] is called the “degree of negative membership of x in A”, and µA (x), ηA (x), νA (x) satisfy µA (x) + ηA (x) + νA (x) ≤ 1,
3
D∗ = {x = (x1 , x2 , x3 )|x ∈ [0, 1] , x1 + x2 + x3 ≤ 1}. From now on, we will assume that if x ∈ D∗ , then x1 , x2 and x3 denote, respectively, the first, the second, and the third component of x, i.e., x = (x1 , x2 , x3 ). The order relation ≤1 on D∗ , defined by x≤1 y ⇔ ((x1 < y1 )∧(x3 ≥ y3 ))∨((x1 = y1 )∧(x3 > y3 ))∨((x1 = y1 )∧(x3 = y3 ) ∧ (x2 ≤ y2 )), for all x ∈ D∗ . Then, (D∗ , ≤1 ) is a complete lattice [3]. We define the first, the second, and the third projection mapping pr1 , pr2 , and pr3 , respectively, on D∗ , defined as pr1 (x) = x1 , pr2 (x) = x2 , and pr3 (x) = x3 , respectively, for all x ∈ D∗ . We denote the the units of D∗ by 1D∗ = (1, 0, 0) and 0D∗ = (0, 0, 1), respectively. Note that if, for x, y ∈ D∗ , neither x≤1 y nor y≤1 x, then x and y are incomparable w.r.t. ≤1 , denoted as x||≤1 y. Using this lattice, we easily see that with every picture fuzzy set A = {(x, µA (x), ηA (x), νA (x))|x ∈ X} corresponds an D∗ -fuzzy set [10], i.e. a mapping
for all x ∈ X. Then, ∀x ∈ X,
is called the “degree of refusal membership of x in A”. Basically, picture fuzzy sets can be adequate in situations when decision makers face their opinions involving their decision making as follows: : yes, abstain, no, and refusal. The voting results are divided into fours accompanied with the number of voters that are ” vote for”,” abstain”,” vote against”, and ”refusal of the voting”. Group ”abstain” means that the voting paper is a white paper rejecting both ”agree” and ” disagree ” for the candidate but still takes the vote. Group ” refusal of voting” is either invalid voting papers or did not take the vote. In this paper, following the research in [4,6], we investigate on negators and De Morgan fuzzy triples on picture fuzzy sets. The definitions and some properties first are given in the paper [9], some properties of involutive fuzzy negators on picture fuzzy sets will be presented in Section 2. Section 3 is devoted for tnorm, t-conorm, and the corresponding DeMorgan picture fuzzy triples. In the conclusion, we will give some discussions on the further research. First we consider some definitions and notations. Let P F S(X) denote the set of all picture fuzzy sets on the universe X. Now, we consider the set D∗ defined by
1 − (µA (x) + ηA (x) + νA (x))
A : X → D∗ : x 7→ (µA (x), ηA (x), νA (x)).
For further usage, we define * * * * * * * * *
L = {x ∈ D∗ |x1 + x2 + x3 = 1}, L∗ = {x ∈ D∗ |x2 = 0}, Y = {x ∈ D∗ |x3 = 0}, Z = {x ∈ D∗ |x1 = 0}, G = {x ∈ D∗ |x2 = 0, x1 + x3 = 1}, X1 = {x ∈ D∗ |x2 = x3 = 0}, X2 = {x ∈ D∗ |x1 = x3 = 0} X3 = {x ∈ D∗ |x1 = x2 = 0}, X1∗ = {x ∈ D∗ |x1 = 0, x2 + x3 = 1}, X3∗ = {x ∈ D∗ |x3 = 0, x1 + x2 = 1}. II.
I NVOLUTIVE PICTURE FUZZY NEGATORS
Picture fuzzy negator concept first is introduced in [9], forms an extension of fuzzy negator and intuitionistic fuzzy negator, and is defined as follows. Definition 2.1 A picture fuzzy negator is a nonincreasing D → D∗ mapping N satisfying ∗
N (0D∗ ) = 1D∗ and N (1D∗ ) = 0D∗ . If N (N (x)) = x, for all x ∈ D∗ , then N is called an involutive picture fuzzy negator. The mapping N0 is defined by N0 (x) = (x3 , 0, x1 ), for all x ∈ D∗ , be a picture fuzzy negator. From now on, if x ∈ D∗ , then let us define x4 = 1 − x1 − x2 − x3 . The mapping NS defined by NS (x) = (x3 , x4 , x1 ), ∗
for all x ∈ D is an involutive negator and is called the standard negator. Now, we shall give some propositions about the new involutive picture fuzzy negators. Proposition 2.2 Let N be an involutive picture fuzzy negator. Then N (X2 ) = X2 , N (0, 0, 0) = (0, 1, 0). Proof. Let N be an involutive picture fuzzy negator and assume that N (0, x2 , 0) ∈ / X2 . Then N (0, x2 , 0) ∈ {(a, 0, 0), (0, 0, c), (a, 0, c), (a, b, c), (0, b, c), (a, b, 0)}, where a, b, c > 0. Assume that N (0, x2 , 0) = (a, 0, 0), where a > 0. Let y, y 0 ∈ D∗ such that y, y 0 ≤1 (a, 0, 0) and y||≤1 y 0 . Since N is involutive and nonincreasing, N (y), N (y 0 )≥1 N (a, 0, 0) = (0, x2 , 0). Then pr3 N (y) = pr3 N (y 0 ) = 0 and N (y), N (y 0 ) are comparable w.r.t. ≤1 . Hence, y, y 0 are comparable w.r.t. ≤1 , a contradiction. Similarly, we obtain contradictions for the five other cases. Thus, N (X2 ) ⊂ X2 . Since N is involutive, it follows easily that N (X2 ) = X2 . Now, we have N (0, 0, 0) = (0, α, 0), where 0 ≤ α ≤ 1. Since N is involutive and nonincreasing, N (0, 1, 0) = (0, β, 0)≤1 N (0, α, 0) = (0, 0, 0),. Then β = 0 and N (0, 0, 0) = (0, 1, 0). Corollary 2.3 Let N be an involutive picture fuzzy negator. Then N (Y ) = Z and N (Z) = Y .
Proof. Assume that N is an involutive picture fuzzy negator. We have to prove that pr3 N (0, x2 , x3 ) = 0 and pr1 N (x1 , x2 , 0) = 0 for all x1 , x2 , x3 ∈ [0, 1]. Since (0, x2 , x3 )≤1 (0, x2 , 0) and N is nonincreasing, N (0, x2 , x3 )≥1 N (0, x2 , 0) = (0, α, 0), α ∈ [0, 1]. Hence, pr3 N (0, x2 , x3 ) = 0. Similarly, pr1 N (x1 , x2 , 0) = 0. We obtain that N (Y ) ⊂ Z and N (Z) ⊂ Y . Since N is involutive, it follows easily that N (Y ) = Z and N (Z) = Y. Proposition 2.4 Let N be an involutive picture fuzzy negator. For all x ∈ D∗ , pr3 N (x1 , 0, 1 − x1 ) = pr3 N (x1 , 1 − x1 , 0) =
pr3 N (x1 , x2 , 0) = pr3 N (x1 , 0, 0)
pr1 N (1 − x3 , 0, x3 ) = pr1 N (0, 1 − x3 , x3 ) =
pr1 N (0, x2 , x3 ) = pr1 N (0, 0, x3 ).
and
Proof. Assume that N is an involutive picture fuzzy negator. If x1 = 1, then the proposition trivially holds. Let x1 ∈ [0, 1), x = (x1 , 0, 1 − x1 ), and x0 = (x1 , x2 , 0). By Corollary 2.3, pr1 N (x0 ) = 0. Assume that pr3 N (x) 6= pr3 N (x0 ), i.e. pr3 N (x) < pr3 N (x0 ). Now, let z = (0, 0, pr3 N (x)) and z 0 = (min(pr1 N (x), 1 − pr3 N (x0 )), 0, pr3 N (x0 )). Since x1 6= 1, N (x), N (x0 ) 6= 0D∗ . Then pr1 N (x) > 0 and pr3 N (x0 ) 6= 1. Thus, z||≤1 z 0 . Since pr3 N (x) < pr3 N (x0 ), N (x0 )
