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The following table lists many specialized symbols commonly used in mathematics. For the HTML codes of mathematical symbols see mathematical HTML.
 Note: This article contains special characters.
Basic mathematical symbols[]
Symbol

Name  Explanation  Examples 

Should be read as  
Category  
=

equality  x = y means x and y represent the same thing or value.  1 + 1 = 2 
is equal to; equals  
everywhere  
≠
<> 
inequation  x ≠ y means that x and y do not represent the same thing or value.  1 ≠ 2 
is not equal to; does not equal  
everywhere  
<
> ≪ ≫ 
strict inequality  x < y means x is less than y. x > y means x is greater than y. x ≪y means x is much less than y. x ≫ y means x is much greater than y. 
3 < 4 5 > 4. 0.003 ≪1,000,000 
is less than, is greater than, is much less than, is much greater than  
order theory  
≤
≥ 
inequality  x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y. 
3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5 
is less than or equal to, is greater than or equal to  
order theory  
∝

proportionality  y ∝ x means that y = kx for some constant k.  if y = 2x, then y ∝ x 
is proportional to  
everywhere  
+

addition  4 + 6 means the sum of 4 and 6.  2 + 7 = 9 
plus  
arithmetic  
disjoint union  A_{1} + A_{2} means the disjoint union of sets A_{1} and A_{2}.  A_{1}={1,2,3,4} ∧ A_{2}={2,4,5,7} ⇒ A_{1} + A_{2} = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}  
the disjoint union of … and …  
set theory  
−

subtraction  9 − 4 means the subtraction of 4 from 9.  8 − 3 = 5 
minus  
arithmetic  
negative sign  −3 means the negative of the number 3.  −(−5) = 5  
negative ; minus  
arithmetic  
settheoretic complement  A − B means the set that contains all the elements of A that are not in B.  {1,2,4} − {1,3,4} = {2}  
minus; without  
set theory  
×

multiplication  3 × 4 means the multiplication of 3 by 4.  7 × 8 = 56 
times  
arithmetic  
Cartesian product  X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y.  {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}  
the Cartesian product of … and …; the direct product of … and …  
set theory  
cross product  u × v means the cross product of vectors u and v  (1,2,5) × (3,4,−1) = (−22, 16, − 2)  
cross  
vector algebra  
÷
/ 
division  6 ÷ 3 or 6/3 means the division of 6 by 3.  2 ÷ 4 = .5 12/4 = 3 
divided by  
arithmetic  
√

square root  √x means the positive number whose square is x.  √4 = 2 
the principal square root of; square root  
real numbers  
complex square root  if z = r exp(iφ) is represented in polar coordinates with π < φ ≤ π, then √z = √r exp(iφ/2).  √(1) = i  
the complex square root of; square root  
complex numbers  
 

absolute value  x means the distance in the real line (or the complex plane) between x and zero.  3 = 3, 5 = 5 i = 1, 3+4i = 5 
absolute value of  
numbers  
!

factorial  n! is the product 1×2×...×n.  4! = 1 × 2 × 3 × 4 = 24 
factorial  
combinatorics  
~

probability distribution  X ~ D, means the random variable X has the probability distribution D.  X ~ N(0,1), the standard normal distribution 
has distribution  
statistics  
⇒
→ ⊃ 
material implication  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 meaning for functions given below. ⊃ may mean the same as ⇒, or it may have the meaning for superset given below. 
x = 2 ⇒ x^{2} = 4 is true, but x^{2} = 4 ⇒ x = 2 is in general false (since x could be −2). 
implies; if .. then  
propositional logic  
⇔
↔ 
material equivalence  A ⇔ B means A is true if B is true and A is false if B is false.  x + 5 = y +2 ⇔ x + 3 = y 
if and only if; iff  
propositional logic  
¬
˜ 
logical negation  The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front. 
¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y) 
not  
propositional logic  
∧

logical conjunction or meet in a lattice  The statement A ∧ B is true if A and B are both true; else it is false.  n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number. 
and  
propositional logic, lattice theory  
∨

logical disjunction or join in a lattice  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. 
or  
propositional logic, lattice theory  
⊕ ⊻ 
exclusive or  The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.  (¬A) ⊕ A is always true, A ⊕ A is always false. 
xor  
propositional logic, Boolean algebra  
∀

universal quantification  ∀ x: P(x) means P(x) is true for all x.  ∀ n ∈ N: n^{2} ≥ n. 
for all; for any; for each  
predicate logic  
∃

existential quantification  ∃ x: P(x) means there is at least one x such that P(x) is true.  ∃ n ∈ N: n is even. 
there exists  
predicate logic  
∃!

uniqueness quantification  ∃! x: P(x) means there is exactly one x such that P(x) is true.  ∃! n ∈ N: n + 5 = 2n. 
there exists exactly one  
predicate logic  
:=
≡ :⇔ 
definition  x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence). P :⇔ Q means P is defined to be logically equivalent to Q. 
cosh x := (1/2)(exp x + exp (−x)) A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) 
is defined as  
everywhere  
{ , }

set brackets  {a,b,c} means the set consisting of a, b, and c.  N = {0,1,2,...} 
the set of ...  
set theory  
{ : }
{  } 
set builder notation  {x : P(x)} means the set of all x for which P(x) is true. {x  P(x)} is the same as {x : P(x)}.  {n ∈ N : n^{2} < 20} = {0,1,2,3,4} 
the set of ... such that ...  
set theory  
Template:0/ {} 
empty set  Template:0/ means the set with no elements. {} means the same.  {n ∈ N : 1 < n^{2} < 4} = Template:0/ 
the empty set  
set theory  
∈
Template:Notin 
set membership  a ∈ S means a is an element of the set S; a Template:Notin S means a is not an element of S.  (1/2)^{−1} ∈ N 2^{−1} Template:Notin N 
is an element of; is not an element of  
everywhere, set theory  
⊆
⊂ 
subset  (subset) A ⊆ B means every element of A is also element of B. (proper subset) A ⊂ B means A ⊆ B but A ≠ B. 
A ∩ B ⊆ A; Q ⊂ R 
is a subset of  
set theory  
⊇
⊃ 
superset  A ⊇ B means every element of B is also element of A. A ⊃ B means A ⊇ B but A ≠ B. 
A ∪ B ⊇ B; R ⊃ Q 
is a superset of  
set theory  
∪

settheoretic union  (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". (inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B. "A or B or both". 
A ⊆ B ⇔ A ∪ B = B (inclusive) 
the union of ... and ...; union  
set theory  
∩

settheoretic intersection  A ∩ B means the set that contains all those elements that A and B have in common.  {x ∈ R : x^{2} = 1} ∩ N = {1} 
intersected with; intersect  
set theory  
\

settheoretic complement  A \ B means the set that contains all those elements of A that are not in B.  {1,2,3,4} \ {3,4,5,6} = {1,2} 
minus; without  
set theory  
( )

function application  f(x) means the value of the function f at the element x.  If f(x) := x^{2}, then f(3) = 3^{2} = 9. 
of  
set theory  
precedence grouping  Perform the operations inside the parentheses first.  (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.  
everywhere  
f:X→Y

function arrow  f: X → Y means the function f maps the set X into the set Y.  Let f: Z → N be defined by f(x) := x^{2}. 
from ... to  
set theory  
o

function composition  fog is the function, such that (fog)(x) = f(g(x)).  if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3). 
composed with  
set theory  
N ℕ

natural numbers  N means {0,1,2,3,...}, but see the article on natural numbers for a different convention.  {a : a ∈ Z} = N 
N  
numbers  
Z ℤ 
integers  Z means {...,−3,−2,−1,0,1,2,3,...}.  {a : a ∈ N} = Z 
Z  
numbers  
Q ℚ 
rational numbers  Q means {p/q : p,q ∈ Z, q ≠ 0}.  3.14 ∈ Q π ∉ Q 
Q  
numbers  
R ℝ 
real numbers  R means the set of real numbers.  π ∈ R √(−1) ∉ R 
R  
numbers  
C ℂ 
complex numbers  C means {a + bi : a,b ∈ R}.  i = √(−1) ∈ C 
C  
numbers  
∞

infinity  ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.  lim_{x→0} 1/x = ∞ 
infinity  
numbers  
pi  π is the ratio of a circle's circumference to its diameter. Its value is 3.1415....  A = πr² is the area of a circle with radius r  
pi  
Euclidean geometry  
 

norm  x is the norm of the element x of a normed vector space.  x+y ≤ x + y 
norm of; length of  
linear algebra  
∑

summation  ∑_{k=1}^{n} a_{k} means a_{1} + a_{2} + ... + a_{n}.  ∑_{k=1}^{4} k^{2} = 1^{2} + 2^{2} + 3^{2} + 4^{2} = 1 + 4 + 9 + 16 = 30 
sum over ... from ... to ... of  
arithmetic  
∏

product  ∏_{k=1}^{n} a_{k} means a_{1}a_{2}···a_{n}.  ∏_{k=1}^{4} (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360 
product over ... from ... to ... of  
arithmetic  
Cartesian product  ∏_{i=0}^{n}Y_{i} means the set of all (n+1)tuples (y_{0},...,y_{n}).  ∏_{n=1}^{3}R = R^{n}  
the Cartesian product of; the direct product of  
set theory  
'

derivative  f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.  If f(x) := x^{2}, then f '(x) = 2x 
… prime; derivative of …  
calculus  
∫

indefinite integral or antiderivative  ∫ f(x) dx means a function whose derivative is f.  ∫x^{2} dx = x^{3}/3 + C 
indefinite integral of …; the antiderivative of …  
calculus  
definite integral  ∫_{a}^{b} f(x) dx means the signed area between the xaxis and the graph of the function f between x = a and x = b.  ∫_{0}^{b} x^{2 } dx = b^{3}/3;  
integral from ... to ... of ... with respect to  
calculus  
∇

gradient  ∇f (x_{1}, …, x_{n}) is the vector of partial derivatives (df / dx_{1}, …, df / dx_{n}).  If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z) 
del, nabla, gradient of  
calculus  
∂

partial derivative  With f (x_{1}, …, x_{n}), ∂f/∂x_{i} is the derivative of f with respect to x_{i}, with all other variables kept constant.  If f(x,y) := x^{2}y, then ∂f/∂x = 2xy 
partial derivative of  
calculus  
boundary  ∂M means the boundary of M  ∂{x : x ≤ 2} = {x : x = 2}  
boundary of  
topology  
⊥

perpendicular  x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.  If l⊥m and m⊥n then l  n. 
is perpendicular to  
geometry  
bottom element  x = ⊥ means x is the smallest element.  ∀x : x ∧ ⊥ = ⊥  
the bottom element  
lattice theory  
⊧

entailment  A ⊧ B means the sentence A entails the sentence B, that is every model in which A is true, B is also true.  A ⊧ A ∨ ¬A 
entails  
model theory  
⊢

inference  x ⊢ y means y is derived from x.  A → B ⊢ ¬B → ¬A 
infers or is derived from  
propositional logic, predicate logic  
◅

normal subgroup  N ◅ G means that N is a normal subgroup of group G.  Z(G) ◅ G 
is a normal subgroup of  
group theory  
/

quotient group  G/H means the quotient of group G modulo its subgroup H.  {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}} 
mod  
group theory  
≈

isomorphism  G ≈ H means that group G is isomorphic to group H  Q / {1, −1} ≈ V, where Q is the quaternion group and V is the Klein fourgroup. 
is isomorphic to  
group theory  
approximately equal  x ≈ y means x is approximately equal to y  π ≈ 3.14159  
is approximately equal to  
everywhere
 
⊗

tensor product  V ⊗ U means the tensor product of V and U.  {1, 2, 3, 4} ⊗ {1,1,2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} 
tensor product of  
linear algebra 
See also[]
 Mathematical alphanumeric symbols
 Physical constants
 Variables commonly used in physics
External links[]
 Jeff Miller: Earliest Uses of Various Mathematical Symbols
 TCAEP  Institute of Physics
 GIF and PNG Images for Math Symbols
Special characters[]
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