Table of mathematical symbols
From Academic Kids

In mathematics, a set of symbols is frequently used in mathematical expressions. As mathematicians are familiar with these symbols, they are not explained each time they are used. So, for mathematical novices, the following table lists many common symbols together with their name, pronunciation and related field of mathematics. Additionally, the third column contains an informal definition, and the fourth column gives a short example.
Be aware that, in some cases, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings. Template:SpecialCharsNote
Basic mathematical symbols
Symbol
 Name  Explanation  Example 

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  
+
 addition  4 + 6 means the sum of 4 and 6.  2 + 7 = 9 
plus  
arithmetic  
−
 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  
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  
⇒
→ ⊃  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 + 5 = 2n 
there exists  
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  
empty set  Template:Unicode means the set with no elements. {} means the same.  {n ∈ N : 1 < n^{2} < 4} = Template:Unicode  
the empty set  
set theory  
∈
∉  set membership  a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S.  (1/2)^{−1} ∈ N 2^{−1} ∉ N 
is an element of; is not an element of  
everywhere, set theory  
⊆
⊂  subset  A ⊆ B means every element of A is also element of B. 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  A ∪ B means the set that contains all the elements from A and also all those from B, but no others.  A ⊆ B ⇔ A ∪ B = B 
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  
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 {lim_{n→∞} a_{n} : ∀ n ∈ N: a_{n} ∈ Q, the limit exists}.  π ∈ R √(−1) ∉ R 
R  
numbers  
C ℂ  complex numbers  C means {a + bi : a,b ∈ R}.  i = √(−1) ∈ C 
C  
numbers  
<
>  strict inequality  x < y means x is less than y. x > y means x is greater than y.  x < y ⇔ y > x 
is less than, is greater than  
partial orders  
≤
≥  inequality  x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y.  x ≥ 1 ⇒ x^{2} ≥ x 
is less than or equal to, is greater than or equal to  
partial orders  
√
 square root  √x means the positive number whose square is x.  √(x^{2}) = x 
the principal square root of; square root  
real 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  
π  pi  π means the ratio of a circle's circumference to its diameter.  A = πr² is the area of a circle with radius r 
pi  
Euclidean geometry  
!
 factorial  n! is the product 1×2×...×n.  4! = 1 × 2 × 3 × 4 = 24 
factorial  
combinatorics  
 
 absolute value  x means the distance in the real line (or the complex plane) between x and zero.  a + bi = √(a^{2} + b^{2}) 
absolute value of  
numbers  
 
 norm  x is the norm of the element x of a normed vector space.  x+y ≤ x + y 
norm of; length of  
functional analysis  
∑
 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  
∫
 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; ∫x^{2} dx = x^{3}/3 
integral from ... to ... of ... with respect to  
calculus  
f '
 derivative  f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there.  If f(x) = x^{2}, then f '(x) = 2x and f ''(x) = 2 
derivative of f; f prime  
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.  
is perpendicular to  
orthogonality  
bottom element  x = ⊥ means x is the smallest element.  
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.  
entails  
propositional logic, predicate logic  
⊢
 inference  x ⊢ y means y is derived from x.  
infers or is derived from  
propositional logic, predicate logic 
If some of these symbols are used in a Wikipedia article that is intended for beginners, it may be a good idea to include a statement like the following, (below the definition of the subject), in order to reach a broader audience:
 ''This article uses [[table of mathematical symbolsmathematical symbols]].''
The article wikipedia:How to edit a page contains information about how to produce these math symbols in Wikipedia articles.
See also:
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
Special characters
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