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Table_of_mathematical_symbols, Displaying_a_formula
The following table lists many specialized symbols commonly used in mathematics.
Basic mathematical symbolsEdit
Σύμβολο  Name  Εξήγηση  Παραδείγματα 

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. (The symbols != and <> are primarily from computer science. They are avoided in mathematical texts.)  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 ≪ 1000000 
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. (The symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.)  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; varies as  
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. ∖ can also be used for settheoretic complement as described below.  {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  
·
 multiplication  3 · 4 means the multiplication of 3 by 4.  7 · 8 = 56 
times  
arithmetic  
dot product  u · v means the dot product of vectors u and v  (1,2,5) · (3,4,−1) = 6  
dot  
vector algebra  
÷
⁄  division  6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.  2 ÷ 4 = .5 12 ⁄ 4 = 3 
divided by  
arithmetic  
±
 plusminus  6 ± 3 means both 6 + 3 and 6  3.  The equation x = 5 ± √4, has two solutions, x = 7 and x = 3. 
plus or minus  
arithmetic  
plusminus  10 ± 2 or eqivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.  If a = 100 ± 1 mm, then a is ≥ 99 mm and ≤ 101 mm.  
plus or minus  
measurement  
∓
 minusplus  6 ± (3 ∓ 5) means both 6 + (3  5) and 6  (3 + 5).  cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y). 
minus or plus  
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(i φ/2).  √(1) = i  
the complex square root of … square root  
complex numbers  
…
 absolute value or modulus  x means the distance along the real line (or across the complex plane) between x and zero.  3 = 3 –5 = 5  i  = 1  3 + 4i  = 5 
absolute value (modulus) of  
numbers  
Euclidean distance  x – y means the Euclidean distance between x and y.  For x = (1,1), and y = (4,5), x – y = √([1–4]^{2} + [1–5]^{2}) = 5  
Euclidean distance between; Euclidean norm of  
Geometry  
Determinant  A means the determinant of the matrix A  
determinant of  
Matrix theory  

 divides  A single vertical bar is used to denote divisibility. ab means a divides b.  Since 15 = 3×5, it is true that 315 and 515. 
divides  
Number Theory  
!
 factorial  n ! is the product 1 × 2× ... × n.  4! = 1 × 2 × 3 × 4 = 24 
factorial  
combinatorics  
T
 transpose  Swap rows for columns  
transpose  
matrix operations  
~
 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  
Row equivalence  A~B means that B can be generated by using a series of elementary row operations on A  
is row equivalent to  
Matrix theory  
⇒
→ ⊃  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, Heyting algebra  
⇔
↔  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. (The symbol ~ has many other uses, so ¬ or the slash notation is preferred.)  ¬(¬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. For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).  n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number. 
and; min  
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. For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).  n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. 
or; max  
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  
direct sum  The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).  Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = ∅)  
direct sum of  
Abstract 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 (Some writers use ≡ to mean 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  
≅
 congruence  △ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.  
is congruent to  
geometry  
≡
 congruence relation  a ≡ b (mod n) means a − b is divisible by n  5 ≡ 11 (mod 3) 
... is congruent to ... modulo ...  
modular arithmetic  
{ , }
 set brackets  {a,b,c} means the set consisting of a, b, and c.  ℕ = { 1, 2, 3, …} 
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} = { 1, 2, 3, 4} 
the set of … such that  
set theory  
∅ { }  empty set  ∅ means the set with no elements. { } means the same.  {n ∈ ℕ : 1 < n^{2} < 4} = ∅ 
the empty set  
set theory  
set membership  a ∈ S means a is an element of the set S; a Πρότυπο:Notin S means a is not an element of S.  (1/2)^{−1} ∈ ℕ 2^{−1} Πρότυπο: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. (Some writers use the symbol ⊂ as if it were the same as ⊆.)  (A ∩ B) ⊆ A ℕ ⊂ ℚ 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. (Some writers use the symbol ⊃ as if it were the same as ⊇.)  (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  
symmetric difference  means the set of elements in exactly one of A or B.  {1,5,6,8} {2,5,8} = {1,2,6}  
symmetric difference  
set theory  
∖
 settheoretic complement  A ∖ B means the set that contains all those elements of A that are not in B. − can also be used for settheoretic complement as described above.  {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.  
parentheses  
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,type 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 { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.  ℕ = {a : a ∈ ℤ, a ≠ 0} 
N  
numbers  
ℤ Z  integers  ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ^{+} means {1, 2, 3, ...} = ℕ.  ℤ = {p, p : p ∈ ℕ} ∪ {0} 
Z  
numbers  
ℚ Q  rational numbers  ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.  3.14000... ∈ 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 + b i : a,b ∈ ℝ}.  i = √(−1) ∈ ℂ 
C  
numbers  
arbitrary constant  C can be any number, most likely unknown; usually occurs when calculating antiderivatives.  if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C, where F'(x) = f(x)  
C  
integral calculus  
𝕂
K  real or complex numbers  K means the statement holds substituting K for R and also for C. 
because and

K  
linear algebra  
∞
 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  
…
 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 
means a_{1} + a_{2} + … + a_{n}. 
= 1^{2} + 2^{2} + 3^{2} + 4^{2}

sum over … from … to … of  
arithmetic  
∏
 product 
means a_{1}a_{2}···a_{n}. 
= (1+2)(2+2)(3+2)(4+2)

product over … from … to … of  
arithmetic  
Cartesian product 
means the set of all (n+1)tuples

 
the Cartesian product of; the direct product of  
set theory  
∐
 coproduct  
coproduct over … from … to … of  
category 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. The dot notation indicates a time derivative. That is .  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  
∮
 contour integral or closed line integral  Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while this formula involves a closed surface integral, the representation describes only the first or initial integration of the volume, over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∰.
This symbol can also frequently be found with a subscript capital letter C, ∮_{C}, denoting that the closed loop integral is around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮_{S}, is used to denote that the integration is over a closed surface.  
contour integral of  
calculus  
∇
 gradient  ∇f (x_{1}, …, x_{n}) is the vector of partial derivatives (∂f / ∂x_{1}, …, ∂f / ∂x_{n}).  If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z) 
del, nabla, gradient of  
vector calculus  
divergence  If , then .  
del dot, divergence of  
vector calculus  
curl  If , then .  
curl of  
vector calculus  
∂
 partial differential  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, d  
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  

 parallel  x  y means x is parallel to y.  If l  m and m ⊥ n then l ⊥ n. 
is parallel to  
geometry  
⊧
 entailment  A ⊧ B means the sentence A entails the sentence B, that is in 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  
quotient set  A/~ means the set of all ~ equivalence classes in A.  If we define ~ by x~y ⇔ xy∈Z, then R/~ = {{x+n : n∈Z} : x ∈ (0,1]}  
mod  
set 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  
~
 same order of magnitude  m ~ n, means the quantities m and n have the general size. (Note that ~ is used for an approximation that is poor, otherwise use ≈ .)  2 ~ 5 8 × 9 ~ 100 but π^{2} ≈ 10 
roughly similar poorly approximates  
Approximation theory
 
〈,〉
(  ) < , > · :  inner product  〈x,y〉 means the inner product of x and y as defined in an inner product space. For spatial vectors, the dot product notation, x·y is common.  The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is: 〈x, y〉 = 2×−1 + 3×5 = 13

inner product of  
linear algebra  
⊗
 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  
*
 convolution  f * g means the convolution of f and g.  
convolution, convoluted with  
functional analysis  
mean  (often read as "x bar") is the mean (average value of ).  .  
overbar, … bar  
statistics  
complex conjugate  is the complex conjugate of z.  
conjugate  
complex numbers  
delta equal to  means equal by definition. When is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡.  .  
equal by definition  
everywhere 
Εσωτερική ΑρθρογραφίαEdit
 Mathematical alphanumeric symbols
 Table of logic symbols
 Mathematical notation
 ISO 3111
 Roman letters used in mathematics
 Greek letters used in mathematics
 Notation in probability
 Physical constants
 Variables commonly used in physics