Table of mathematical symbols From Wikipedia, the free encyclopedia For the HTML codes of mathematical symbols see mathematical HTML. Note: This article contains ...
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Table of mathematical symbols From Wikipedia, the free encyclopedia
For the HTML codes of mathematical symbols see mathematical HTML. Note: This article contains special characters. The following table lists many specialized symbols commonly used in mathematics.
Basic mathematical symbols Name
Symbol
Read as
Explanation
Examples
Category equality
=
is equal to; equals
x = y means x and y represent the same thing or value.
1+1=2
everywhere
≠ !=
≪ ≫
is less than, is greater than, is much less than, is much greater than
x ≫ y means x is much greater than y. order theory inequality
≤ y means x is greater 3 4. x ≪ y means x is much 0.003 ≪ 1000000 less than y.
is less than or
x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to
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≥ >=
equal to, is greater than or equal to
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y.
(The symbols = are primarily from 5 ≥ 4 and 5 ≥ 5 computer science. They order theory are avoided in mathematical texts.)
proportionality
∝
is proportional y ∝ x means that y = kx if y = 2x, then y ∝ x to; varies as for some constant k. everywhere addition plus
4 + 6 means the sum of 2+7=9 4 and 6.
arithmetic
+
disjoint union the disjoint union of ... and ...
A1 + A2 means the A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒ disjoint union of sets A1 A + A = {(1,1), (2,1), (3,1), (4,1), (2,2), 1 2 and A2. (4,2), (5,2), (7,2)}
set theory subtraction minus
9 − 4 means the 8−3=5 subtraction of 4 from 9.
arithmetic negative sign
−
negative ; minus
−3 means the negative of the number 3.
−(−5) = 5
arithmetic settheoretic complement
A − B means the set that contains all the minus; without elements of A that are set theory not in B.
{1,2,4} − {1,3,4} = {2}
multiplication
3 × 4 means the times multiplication of 3 by arithmetic 4.
7 × 8 = 56
Cartesian product
×
X×Y means the set of the Cartesian all ordered pairs with product of ... the first element of each {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)} and ...; the direct pair selected from X product of ... and the second element and ... selected from Y. set theory
cross product
u × v means the cross product of vectors u vector algebra and v cross
(1,2,5) × (3,4,−1) = (−22, 16, − 2)
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multiplication
3 · 4 means the multiplication of 3 by arithmetic 4.
times
·
dot product
u · v means the dot product of vectors u vector algebra and v dot
÷
6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.
2 ÷ 4 = .5 12 ⁄ 4 = 3
arithmetic plusminus plus or minus
±
(1,2,5) · (3,4,−1) = 6
division divided by
⁄
7 · 8 = 56
arithmetic
6 ± 3 means both 6 + 3 The equation x = 5 ± √4, has two and 6  3. solutions, x = 7 and x = 3.
plusminus
10 ± 2 or eqivalently 10 If a = 100 ± 1 mm, then a is ≥ 99 mm ± 20% means the range and ≤ 101 mm. measurement from 10 − 2 to 10 + 2.
plus or minus
∓
minusplus
6 ± (3 5) means both 6 + (3  5) and 6  (3 + cos(x ± y) = cos(x) cos(y) arithmetic 5).
minus or plus
sin(x) sin(y).
square root the principal square root of; square root
√x means the positive number whose square is √4 = 2 x.
real numbers
√
…
complex square root
if z = r exp(iφ) is the complex represented in polar square root of … coordinates with π < φ √(1) = i ≤ π, then √z = √r exp square root (i φ/2). complex numbers
absolute value or x means the distance modulus along the real line (or absolute value across the complex (modulus) of plane) between x and zero. numbers
3 = 3 –5 = 5 i=1  3 + 4i  = 5
Euclidean distance Euclidean distance
x – y means the
For x = (1,1), and y = (4,5),
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between; Euclidean distance Euclidean norm between x and y. of
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x – y = √([1–4]2 + [1–5]2) = 5
Geometry Determinant
A means the determinant of determinant of the Matrix theory matrix A A single vertical bar is used to denote divides divisibility. Number Theory ab means a divides b.
divides

Since 15 = 3×5, it is true that 315 and 515.
factorial
!
factorial
n ! is the product 1 × 2× ... × n.
4! = 1 × 2 × 3 × 4 = 24
combinatorics transpose
T
transpose matrix operations
Swap rows for columns Aij = (AT)ji
probability distribution
~
⇒ → ⊃ ⇔
X ~ D, means the random variable X has has distribution the probability statistics distribution D.
X ~ N(0,1), the standard normal distribution
Row equivalence A~B means that B can is row equivalent be generated by using a to series of elementary Matrix theory row operations on A material implication
implies; if … then
A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B. → may mean the same as ⇒, or it may have the x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = meaning for functions 2 is in general false (since x could be −2). given below.
propositional ⊃ may mean the same logic, Heyting as ⇒, or it may have the algebra meaning for superset given below. material equivalence
A ⇔ B means A is true if B is true and A is if and only if; iff false if B is false.
x + 5 = y +2 ⇔ x + 3 = y
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↔
propositional logic logical negation
¬ ˜
∧
not
The statement ¬A is true if and only if A is false. A slash placed through another operator is the ¬(¬A) ⇔ A same as "¬" placed in x ≠ y ⇔ ¬(x = y) front.
propositional (The symbol ~ has logic many other uses, so ¬ or the slash notation is preferred.) The statement A ∧ B is logical conjunction or true if A and B are both meet in a lattice true; else it is false. and; min
For functions A(x) and propositional B(x), A(x) ∧ B(x) is logic, lattice used to mean min(A(x), theory B(x)). logical disjunction or join in a lattice
∨
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or; max
The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.
n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.
n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number.
For functions A(x) and propositional B(x), A(x) ∨ B(x) is logic, lattice theory used to mean max(A (x), B(x)). exclusive or
The statement A ⊕ B is true when either A or (¬A) ⊕ A is always true, A ⊕ A is propositional B, but not both, are always false. logic, Boolean true. A B means the algebra same. xor
⊕
The direct sum is a special way of combining several one modules into one direct sum of general module (the symbol ⊕ is used, is only for logic). Abstract algebra
direct sum
universal quantification
Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = ∅)
∀n∈
: n 2 ≥ n.
for all; for any; ∀ x: P(x) means P(x) is
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∀ ∃
for each
true for all x.
predicate logic existential quantification there exists predicate logic
∃!
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uniqueness quantification there exists exactly one
∃ x: P(x) means there is at least one x such that ∃ n ∈ P(x) is true.
∃! x: P(x) means there is exactly one x such that P(x) is true.
∃! n ∈
: n is even.
: n + 5 = 2n.
predicate logic
:= ≡ :⇔
definition
is defined as
x := y or x ≡ y means x is defined to be another name for y (Some writers use ≡ to mean congruence).
cosh x := (1/2)(exp x + exp (−x)) A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
P :⇔ Q means P is everywhere defined to be logically equivalent to Q. △ABC △DEF means triangle ABC is is congruent to congruent to (has the same measurements as) geometry triangle DEF.
congruence
congruence relation
≡
... is congruent a ≡ b (mod n) means a to ... modulo ... − b is divisible by n modular arithmetic
5 ≡ 11 (mod 3)
set brackets
{,} {:} {} ∅
{a,b,c} means the set the set of … consisting of a, b, and set theory c.
set builder notation the set of … such that
= { 1, 2, 3, …}
{x : P(x)} means the set of all x for which P(x) {n ∈ is true. {x  P(x)} is the same as {x : P(x)}.
: n2 < 20} = { 1, 2, 3, 4}
∅ means the set with no elements. { } means {n ∈ the same.
: 1 < n2 < 4} = ∅
set theory empty set the empty set
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{} ∈ ∉ ⊆
set theory set membership is an element of; a ∈ S means a is an (1/2)−1 ∈ is not an element element of the set S; a ∉ of S means a is not an 2−1 ∉ everywhere, set element of S. theory (subset) A ⊆ B means every element of A is also element of B.
subset
is a subset of
⊂
⊇
superset
(A ∩ B) ⊆ A ⊂ ⊂
A ⊇ B means every element of B is also element of A. A ⊃ B means A ⊇ B but (A ∪ B) ⊇ B A ≠ B.
⊃
(Some writers use the set theory symbol ⊃ as if it were the same as ⊇.) settheoretic union
∪
(proper subset) A ⊂ B means A ⊆ B but A ≠ B.
(Some writers use the set theory symbol ⊂ as if it were the same as ⊆.)
is a superset of
⊃
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the union of … and
(exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both. "A or B, but not both."
A ⊆ B ⇔ (A ∪ B) = B (inclusive)
(inclusive) A ∪ B means the set that contains all union the elements from A, or all the elements from B, or all the elements from set theory both A and B. "A or B or both".
settheoretic intersection
A ∩ B means the set that contains all those intersected with; elements that A and B
{x ∈
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: x2 = 1} ∩
= {1}
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∩
intersect
have in common.
set theory symmetric difference
∆
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symmetric difference
A∆B means the set of elements in exactly one {1,5,6,8} ∆ {2,5,8} = {1,2,6} of A or B.
set theory settheoretic complement
A B means the set that contains all those minus; without elements of A that are set theory not in B.
function application of
()
{1,2,3,4}
{3,4,5,6} = {1,2}
f(x) means the value of the function f at the If f(x) := x2, then f(3) = 32 = 9. element x.
set theory precedence grouping parentheses
Perform the operations inside the parentheses (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. first.
everywhere function arrow
f:X→Y o
f: X → Y means the function f maps the set set theory,type X into the set Y. theory from … to
Let f:
→
be defined by f(x) := x2.
function composition
fog is the function, such if f(x) := 2x, and g(x) := x + 3, then (fog) composed with that (fog)(x) = f(g(x)). (x) = 2(x + 3). set theory
natural numbers N
N
numbers integers Z
Z
numbers
N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.
= {a : a ∈
means {..., −3, −2, −1, 0, 1, 2, 3, ...} and + means {1, 2, 3, ...} = .
= {p, p : p ∈
rational numbers Q
Q
means {p/q : p ∈ q ∈ }.
,
, a ≠ 0}
} ∪ {0}
3.14000... ∈ π
numbers
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real numbers R
R
means the set of real numbers.
numbers complex numbers C
means {a + b i : a,b ∈ }.
π∈ √(−1)
i = √(−1) ∈
numbers
C
C can be any number, most likely unknown; usually occurs when C calculating integral calculus antiderivatives.
arbitrary constant
real or complex numbers
K
K
if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C, where F'(x) = f(x)
because K means the statement holds substituting K for R and also for C. and
linear algebra
.
∞ is an element of the extended number line infinity that is greater than all real numbers; it often numbers occurs in limits.
infinity
∞
limx 0 1/x = ∞ →
norm
…
norm of length of
 x  is the norm of the element x of a normed vector space.
 x + y  ≤  x  +  y 
linear algebra summation
∑
sum over … from … to … of arithmetic
means a1 + a2 + … + an.
= 1 2 + 2 2 + 3 2 + 42 = 1 + 4 + 9 + 16 = 30
product
∏
product over … from … to … of
means
a a ···a . arithmetic 1 2 n
= (1+2)(2+2)(3+2)(4+2) = 3 × 4 × 5 × 6 = 360
Cartesian product
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the Cartesian product of; the direct product of
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means the set of all (n+1)tuples
set theory (y0, …, yn). coproduct coproduct over … from … to … of category theory derivative
′
… prime derivative of
•
f ′(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. The dot notation indicates a time derivative. That is
If f(x) := x2, then f ′(x) = 2x
.
calculus indefinite integral or antiderivative indefinite integral of
∫
∫ f(x) dx means a function whose derivative is f.
∫x2 dx = x3/3 + C
the antiderivative of calculus definite integral
∫ab f(x) dx means the integral from … signed area between the to … of … with xaxis and the graph of ∫0b x2 dx = b3/3; respect to the function f between calculus x = a and x = b. ∇f (x1, …, xn) is the vector of partial If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, del, nabla, f x derivatives ( / , …, ∂ ∂ 1 gradient of 2z) calculus ∂f / ∂xn).
gradient
∇
partial differential
∂
With f (x1, …, xn), ∂f/∂x is the derivative i
2 of f with respect to xi, If f(x,y) := x y, then ∂f/∂x = 2xy partial, d with all other variables calculus kept constant.
boundary
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∂M means the boundary ∂{x : x ≤ 2} = {x : x = 2} topology of M
boundary of perpendicular
⊥
x ⊥ y means x is is perpendicular perpendicular to y; or to more generally x is geometry orthogonal to y.
If l ⊥ m and m ⊥ n then l  n.
bottom element the bottom element
x = ⊥ means x is the smallest element.
∀x : x ∧ ⊥ = ⊥
lattice theory parallel

is parallel to
x  y means x is parallel If l  m and m ⊥ n then l ⊥ n. to y.
geometry A B means the sentence A entails the A entails sentence B, that is every model in which A model theory is true, B is also true.
entailment
A ∨ ¬A
inference infers or is derived from
x y means y is derived from x. propositional logic, predicate logic
A→B
¬B → ¬A
normal subgroup is a normal subgroup of
N G means that N is a normal subgroup of group G.
Z(G)
G
group theory quotient group
G/H means the quotient {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, of group G modulo its b}, {a, b+a}, {2a, b+2a}} group theory subgroup H. mod
/
quotient set
A/~ means the set of all If we define ~ by x~y ⇔ xy∈Z, then mod ~ equivalence classes in R/~ = {{x+n : n∈Z} : x ∈ (0,1]} A. set theory
isomorphism
G ≈ H means that is isomorphic to group G is isomorphic group theory to group H
≈
Q / {1, −1} ≈ V, where Q is the quaternion group and V is the Klein fourgroup.
approximately equal
x ≈ y means x is is approximately approximately equal to π ≈ 3.14159 equal to y everywhere
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same order of magnitude
~
m ~ n, means the roughly similar quantities m and n have 2 ~ 5 the general size. poorly 8 × 9 ~ 100 approximates (Note that ~ is used for Approximation an approximation that but π2 ≈ 10 theory is poor, otherwise use ≈ .)
〈,〉 inner product 〈x,y〉 means the inner product of x and y The standard inner product between two as defined in an inner vectors x = (2, 3) and y = (−1, 5) is: product space. 〈x, y〉 = 2×−1 + 3×5 = 13
()
inner product of
·
For spatial vectors, the A:B = ∑ AijBij dot product notation, x·y is common. i,j For matricies, the colon notation may be used.
vector algebra
: tensor product tensor product of
V U means the tensor {1, 2, 3, 4} {1,1,2} = product of V and U. {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
linear algebra
*
convolution convolution mean overbar statistics
f * g means the convolution of f and g. is the mean (average value of xi).
means equal by definition. When is used, equality is not true generally, but equal by rather equality is true definition under certain assumptions that are everywhere taken in context.
.
delta equal to
.
See also
Mathematical alphanumeric symbols
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Table of logic symbols Physical constants Variables commonly used in physics ISO 3111
External links
Jeff Miller: Earliest Uses of Various Mathematical Symbols (http://members.aol.com/jeff570/mathsym.html) TCAEP  Institute of Physics (http://www.tcaep.co.uk/science/symbols/maths.htm) GIF and PNG Images for Math Symbols (http://us.metamath.org/symbols/symbols.html)
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