Table of mathematical symbols

Encyclopedia : T : TA : TAB : Table of mathematical symbols

Note: This article contains .

The following table lists many specialized symbols commonly used in mathematics. For the HTML codes of mathematical symbols see mathematical HTML.

Contents

1 Basic mathematical symbols
2 See also
3 External links
4 Special characters

Basic mathematical symbols

∧ A₂ = ⇒
A₁ + A₂ = |- |align=center|the disjoint union of ... and ... |- |align=right|set theory

|- | rowspan=9 bgcolor=#d0f0d0 align=center|

−

|- | rowspan=9 bgcolor=#d0f0d0 align=center|

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

÷

/

||division | rowspan=3|6 ÷ 3 or 6/3 means the division of 6 by 3. | rowspan=3|2 ÷ 4 = .5

12/4 = 3 |- |align=center|divided by |- |align=right|arithmetic

|- | rowspan=6 bgcolor=#d0f0d0 align=center|

√

||square root | rowspan=3|√x means the positive number whose square is x. | rowspan=3|√4 = 2 |- |align=center|the principal square root of; square root |- |align=right|real numbers

|- ||complex square root | rowspan=3| if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(iφ/2). | rowspan=3|√(-1) = i |- |align=center|the complex square root of; square root |- |align=right|complex numbers

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

| |

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

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

||factorial | rowspan=3|n! is the product 1 × 2× ... × n. | rowspan=3|4! = 1 × 2 × 3 × 4 = 24 |- |align=center|factorial |- |align=right|combinatorics

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

||probability distribution | rowspan=3| X ~ D, means the random variable X has the probability distribution D. | rowspan=3|''X ~ N(0,1), the standard normal distribution |- |align=center|has distribution |- |align=right|statistics

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

⇒

→

⊃

||material implication | rowspan=3|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. | rowspan=3|x = 2 ⇒ x² = 4 is true, but x² = 4 ⇒ x = 2 is in general false (since x could be −2). |- |align=center|implies; if .. then |- |align=right|propositional logic

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

⇔

↔

||material equivalence | rowspan=3|A ⇔ B means A is true if B is true and A is false if B is false. | rowspan=3|x + 5 = y +2 ⇔ x + 3 = y |- |align=center|if and only if; iff |- |align=right|propositional logic

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

¬

˜

||logical negation | rowspan=3|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. | rowspan=3|¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y) |- |align=center|not |- |align=right|propositional logic

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

∧

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

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

∨

||logical disjunction or join in a lattice | rowspan=3|The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. | rowspan=3|n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. |- |align=center|or |- |align=right|propositional logic, lattice theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

⊕

⊻

||exclusive or | rowspan=3| The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. | rowspan=3| (¬A) ⊕ A is always true, A ⊕ A is always false. |- |align=center|xor |- |align=right|propositional logic, Boolean algebra

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

∀

||universal quantification | rowspan=3|∀ x: P(x) means P(x) is true for all x. | rowspan=3|∀ n ∈ N: n² ≥ n. |- |align=center|for all; for any; for each |- |align=right|predicate logic

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

∃

||existential quantification | rowspan=3|∃ x: P(x) means there is at least one x such that P(x) is true. | rowspan=3|∃ n ∈ N: n is even. |- |align=center|there exists |- |align=right|predicate logic

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

∃!

||uniqueness quantification | rowspan=3|∃! x: P(x) means there is exactly one x such that P(x) is true. | rowspan=3|∃! n ∈ N: n + 5 = 2n. |- |align=center|there exists exactly one |- |align=right|predicate logic

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

:=

≡

:⇔

||definition | rowspan=3|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. | rowspan=3|cosh x := (1/2)(exp x + exp (−x))

A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) |- |align=center|is defined as |- |align=right|everywhere

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

||set brackets | rowspan=3| means the set consisting of a, b, and c. | rowspan=3|N = |- |align=center|the set of ... |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

||set builder notation | rowspan=3| means the set of all x for which P(x) is true. is the same as . | rowspan=3| = |- |align=center|the set of ... such that ... |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

∅

||empty set | rowspan=3|∅ means the set with no elements. means the same. | rowspan=3| = ∅ |- |align=center| the empty set |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

∈

∉

||set membership | rowspan=3|a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S. | rowspan=3|(1/2)⁻¹ ∈ N

2⁻¹ ∉ N |- |align=center|is an element of; is not an element of |- |align=right|everywhere, set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

⊆

⊂

||subset | rowspan=3|(subset) A ⊆ B means every element of A is also element of B.

(proper subset) A ⊂ B means A ⊆ B but A ≠ B. | rowspan=3|A ∩ B ⊆ A; Q ⊂ R |- |align=center|is a subset of |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

⊇

⊃

||superset | rowspan=3|A ⊇ B means every element of B is also element of A.

A ⊃ B means A ⊇ B but A ≠ B. | rowspan=3|A ∪ B ⊇ B; R ⊃ Q |- |align=center|is a superset of |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

∪

||set-theoretic union | rowspan=3|(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". | rowspan=3|A ⊆ B ⇔ A ∪ B = B (inclusive) |- |align=center|the union of ... and ...; union |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

∩

||set-theoretic intersection | rowspan=3|A ∩ B means the set that contains all those elements that A and B have in common. | rowspan=3| ∩ N = |- |align=center|intersected with; intersect |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

||set-theoretic complement | rowspan=3|A \ B means the set that contains all those elements of A that are not in B. | rowspan=3| \ = |- |align=center|minus; without |- |align=right|set theory

|- | rowspan=6 bgcolor=#d0f0d0 align=center|

( )

||function application | rowspan=3|f(x) means the value of the function f at the element x. | rowspan=3|If f(x) := x², then f(3) = 3² = 9. |- |align=center|of |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

f:X→Y

||function arrow | rowspan=3|f: X → Y means the function f maps the set X into the set Y. | rowspan=3|Let f: Z → N be defined by f(x) := x². |- |align=center|from ... to |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

||function composition | rowspan=3|fog is the function, such that (fog)(x) = f(g(x)). | rowspan=3|if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3). |- |align=center|composed with |- |align=right|set theory

|- | rowspan=3 bgcolor=#d0f0d0 align=center|

ℕ

||natural numbers | rowspan=3|N means , but see the article on natural numbers for a different convention. | rowspan=3|a| : a ∈ Z} = N>

numbers

ℤ

integers

Z means .

= Z

numbers

ℚ

rational numbers

Q means .

3.14 ∈ Q

π ∉ Q

numbers

ℝ

real numbers

R means the set of real numbers.

π ∈ R

√(−1) ∉ R

numbers

ℂ

complex numbers

C means .

i = √(−1) ∈ 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

integral calculus

∞

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

Euclidean geometry

norm

is the norm of the element x of a normed vector space.

x+y

≤

norm of; length of

linear algebra

∑

summation

[\sum_^] means a₁ + a₂ + ... + a_n.

[\sum_^] = 1² + 2² + 3² + 4²

:= 1 + 4 + 9 + 16 = 30

sum over ... from ... to ... of

arithmetic

∏

product

[\prod_^na_k] means a₁a₂···a_n.

[\prod_^4(k+2)] = (1+2)(2+2)(3+2)(4+2)

:= 3 × 4 × 5 × 6 = 360

product over ... from ... to ... of

arithmetic

Cartesian product

[\prod_^] means the set of all (n+1)-tuples

:(y₀,...,y_n).

[\prod_^} = \mathbb\times\mathbb\times\mathbb = \mathbb^3]

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) := 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

∫_a^b f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.

∫₀^b x² dx = b³/3;

integral from ... to ... of ... with respect to

calculus

∇

gradient

∇f (x₁, …, x_n) is the vector of partial derivatives (df / dx₁, …, df / dx_n).

If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)

del, nabla, gradient of

calculus

∂

partial derivative

With f (x₁, …, 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) := x²y, then ∂f/∂x = 2xy

partial derivative of

calculus

boundary

∂M means the boundary of M

∂ =

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

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.

/ =

mod

group theory

quotient set

A/~ means the set of all ~ equivalence classes in A.

set theory

≈

isomorphism

G ≈ H means that group G is isomorphic to group H

Q / ≈ 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

⊗

tensor product

V ⊗ U means the tensor product of V and U.

⊗ =

tensor product of

linear algebra

External links

Special characters

Technical note: Due to technical limitations, many computers cannot display some of the special characters in this article. Such characters may be rendered as boxes, question marks, or other nonsense symbols, depending on your browser, operating system, and installed fonts. Even if you have ensured that your browser is interpreting the article as UTF-8 encoded and you have installed a font that supports a wide range of Unicode, such as Code2000, Arial Unicode MS, Lucida Sans Unicode or one of the free Unicode fonts, you may still need to use a different browser, as browser capabilities in this regard tend to vary.

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.

Table of mathematical symbols

Basic mathematical symbols

See also

External links

Special characters