Symbol
| Name |
Explanation |
Examples |
| 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 |
A1 + A2 means the disjoint union of sets A1 and A2. |
A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒
A1 + A2 = {(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 |
| set-theoretic complement |
A − B means the set that contains all the elements of A that are not in B.
∖ can also be used for set-theoretic 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 |
|
±
|
plus-minus |
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 |
| plus-minus |
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 |
|
∓
|
minus-plus |
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 |
<math>\begin{vmatrix}
1&2 \\ 2&4 \\
\end{vmatrix} = 0</math>
|
| determinant of |
| Matrix theory |
|
|
|
divides |
A single vertical bar is used to denote divisibility.
a|b means a divides b. |
Since 15 = 3×5, it is true that 3|15 and 5|15. |
| 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 |
<math>A_{ij} = (A^T)_{ji}</math> |
| 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 |
<math>\begin{bmatrix}
1&2 \\ 2&4 \\
\end{bmatrix} \sim \begin{bmatrix}
1&2 \\ 0&0 \\
\end{bmatrix}</math>
|
| 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 ⇒ x2 = 4 is true, but x2 = 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: n2 ≥ 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 : n2 < 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 < n2 < 4} = ∅ |
| the empty set |
| set theory |
|
|
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.
(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 |
|
∪
|
set-theoretic 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 |
|
∩
|
set-theoretic intersection |
A ∩ B means the set that contains all those elements that A and B have in common. |
{x ∈ ℝ : x2 = 1} ∩ ℕ = {1} |
| intersected with; intersect |
| set theory |
|
<math>\Delta</math>
|
symmetric difference |
<math> A\Delta B</math> means the set of elements in exactly one of A or B. |
{1,5,6,8} <math>\Delta</math> {2,5,8} = {1,2,6} |
| symmetric difference |
| set theory |
|
∖
|
set-theoretic complement |
A ∖ B means the set that contains all those elements of A that are not in B.
− can also be used for set-theoretic 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) := x2, then f(3) = 32 = 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) := x2. |
| 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. |
- <math>x^2\in\mathbb{C}\,\forall x\in \mathbb{K}</math>
because
- <math>x^2\in\mathbb{C}\,\forall x\in \mathbb{R}</math>
and
- <math>x^2\in\mathbb{C}\,\forall x\in \mathbb{C}</math>.
|
| K |
| linear algebra |
|
∞
|
infinity |
∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. |
<math>\lim_{x\to 0} \frac{1}{|x|} = \infty</math> |
| 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 |
<math>\sum_{k=1}^{n}{a_k}</math> means a1 + a2 + … + an.
|
<math>\sum_{k=1}^{4}{k^2}</math> = 12 + 22 + 32 + 42
-
- = 1 + 4 + 9 + 16 = 30
|
| sum over … from … to … of |
| arithmetic |
|
∏
|
product |
<math>\prod_{k=1}^na_k</math> means a1a2···an.
|
<math>\prod_{k=1}^4(k+2)</math> = (1+2)(2+2)(3+2)(4+2)
-
- = 3 × 4 × 5 × 6 = 360
|
| product over … from … to … of |
| arithmetic |
| Cartesian product |
<math>\prod_{i=0}^{n}{Y_i}</math> means the set of all (n+1)-tuples
-
- (y0, …, yn).
|
<math>\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3</math>
|
| 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 <math>\dot{x}(t)=\frac{\partial}{\partial t}x(t)</math>.
|
If f(x) := x2, then f ′(x) = 2x |
… prime
derivative of |
| calculus |
|
∫
|
indefinite integral or antiderivative |
∫ f(x) dx means a function whose derivative is f. |
∫x2 dx = x3/3 + C |
indefinite integral of
the antiderivative of |
| calculus |
| definite integral |
∫ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. |
∫0b x2 dx = b3/3; |
| 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 these formulas involve a closed surface integral, the representations describe only the first 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 ∰.
The contour integral can also frequently be found with a subscript capital letter C, ∮C, denoting that a closed loop integral is, in fact, 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 (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn). |
If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z) |
| del, nabla, gradient of |
| vector calculus |
| divergence |
<math> \nabla \cdot \vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z} </math> |
If <math> \vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k} </math>, then <math> \nabla \cdot \vec v = 3y + 2yz </math>. |
| del dot, divergence of |
| vector calculus |
| curl |
<math> \nabla \times \vec v = \left( {\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \mathbf{i} + \left( {\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \mathbf{j} + \left( {\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \mathbf{k} </math> |
If <math> \vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k} </math>, then <math> \nabla\times\vec v = -y^2\mathbf{i} - 3x\mathbf{k} </math>. |
| curl of |
| vector calculus |
|
∂
|
partial differential |
With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant. |
If f(x,y) := x2y, 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 ⇔ x-y∈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 four-group. |
| 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 π2 ≈ 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.
For matricies, the colon notation may be used.
|
The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is:
〈x, y〉 = 2×−1 + 3×5 = 13
<math>A:B = \sum_{i,j} A_{ij}B_{ij}</math>
|
| 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. |
<math>(f * g )(t) = \int f(\tau) g(t - \tau)\, d\tau</math> |
| convolution, convoluted with |
| functional analysis |
|
<math>\bar{x}</math>
|
mean |
<math>\bar{x}</math> (often read as "x bar") is the mean (average value of <math>x_i</math>). |
<math>x = \{1,2,3,4,5\}; \bar{x} = 3</math>. |
| overbar, … bar |
| statistics |
|
<math> \overline{z} </math>
|
complex conjugate |
<math> \overline{z} </math> is the complex conjugate of z. |
<math> \overline{3+4i} = 3-4i </math> |
| conjugate |
| complex numbers |
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<math>\triangleq</math>
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delta equal to |
<math>\triangleq</math> means equal by definition. When <math>\triangleq</math> is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡. |
<math>p(x_1,x_2,...,x_n) \triangleq \prod_{i=1}^n p(x_i | x_{\pi_i})</math>. |
| equal by definition |
| everywhere |