User:H2g2bob/sandbox

This is a sandbox for Table of mathematical symbols

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The following table lists many specialized symbols commonly used in mathematics.

Equality, inequality

symbol	symbol name	reads as	used in	brief description	examples
=	equality	is equal to; equals	everywhere	x = y means x and y represent the same thing or value. In computing, multiple equals signs are often used for comparison, while single equals signed are used for assignment.	1 + 1 = 2
≠ <> !=	inequation	is not equal to; does not equal	everywhere	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
>	strict inequality	is greater than	order theory	x > y means x is greater than y.	2 > 1
<	strict inequality	is less than	order theory	x < y means x is less than y.	1 < 2
>>	strict inequality	is much greater than	order theory	x >> y means x is much greater than y.	1000 >> 1
<<	strict inequality	is much less than	order theory	x >> y means x is much less than y.	1 << 1000
≥	inequality	is greater than or equal to	order theory	x ≥ y means x is greater than or equal to y.	2 ≥ 1
≤	inequality	is less than or equal to	order theory	x ≤ y means x is less than or equal to y.	1 ≤ 1
∝	proportionality	is proportional to	everywhere	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x
symbol	symbol name	reads as	used in	brief description	examples

Basic operators

symbol	symbol name	reads as	used in	brief description	examples
+	addition	plus	arithmetic	4 + 6 means the sum of 4 and 6.	2 + 7 = 9
−	subtraction	minus	arithmetic	9 − 4 means the subtraction of 4 from 9.	8 − 3 = 5
−	negative sign	negative ; minus	arithmetic	− 3 means the negative of the number 3.	−(−5) = 5
× * ·	multiplication	times	arithmetic	3 × 4 means the multiplication of 3 by 4.	7 × 8 = 56
÷ ⁄	division	divided by	arithmetic	6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.	2 ÷ 4 = .5 12 ⁄ 4 = 3
symbol	symbol name	reads as	used in	brief description	examples

Vector algebra

symbol	symbol name	reads as	used in	brief description	examples
× ∧	cross product	cross	vector algebra	u × v means the cross product of vectors u and v	(1,2,5) × (3,4,−1) = (−22, 16, − 2)
·	dot product	dot	vector algebra	u · v means the dot product of vectors u and v	(1,2,5) · (3,4,−1) = 6
symbol	symbol name	reads as	used in	brief description	examples

Set theory

symbol	symbol name	reads as	used in	brief description	examples
+	disjoint union	the disjoint union of ... and ...	set theory	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)}
−	set-theoretic complement	minus; without	set theory	A − B means the set that contains all the elements of A that are not in B.	{1,2,4} − {1,3,4} = {2}
×	Cartesian product	the Cartesian product of ... and ...; the direct product of ... and ...	set theory	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)}
symbol	symbol name	reads as	used in	brief description	examples

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\pm

\mp

||plus-minus | rowspan=3|6 $\pm$ 3 means both 6 + 3 and 6 - 3.
6 $\pm$ 3 $\mp$ 5 means both 6 + 3 - 5 and 6 - 3 + 5. | rowspan=3|6 $\pm$ 3 = 9 or 3
6 $\pm$ 3 $\mp$ 5 = 4 or 8 |- |align=center|plus-minus; plus-or-minus
minus-plus; minus-or-plus |- |align=right|arithmetic

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√

||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(i φ/2). | rowspan=3|√(-1) = i |- |align=center|the complex square root of …

square root |- |align=right|complex numbers

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|…|

||absolute value | rowspan=3| |x| means the distance along the real line (or across 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

|- ||Euclidean distance | rowspan=3| |x – y| means the Euclidean distance between x and y. | rowspan=3| For x = (1,1), and y = (4,5),
|x – y| = √([1–4]² + [1–5]²) = 5 |- |align=center|Euclidean distance between; Euclidean norm of |- |align=right|Geometry

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|

||divides | rowspan=3| A single vertical bar is used to denote divisibility.
a|b means a divides b. | rowspan=3| Since 15 = 3×5, it is true that 3|15 and 5|15. |- |align=center|divides |- |align=right|Number Theory

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!

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

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T

||transpose | rowspan=3| Swap rows for columns | rowspan=3| $A_{ij}=(A^{T})_{ji}$ |- |align=center|transpose |- |align=right|matrix operations

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~

||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

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⇒

→

⊃

||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

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⇔

↔

||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

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¬

˜

||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.

(The symbol ~ has many other uses, so ¬ or the slash notation is preferred.) | rowspan=3|¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y) |- |align=center|not |- |align=right|propositional logic

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\wedge

||logical conjunction or meet in a lattice | rowspan=3|The statement A $\wedge$ B is true if A and B are both true; else it is false.

For functions A(x) and B(x), A(x) $\wedge$ B(x) is used to mean min(A(x), B(x)). | rowspan=3|n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number. |- |align=center|and; min |- |align=right|propositional logic, lattice theory

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∨

||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.

For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)). | rowspan=3|n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. |- |align=center|or; max |- |align=right|propositional logic, lattice theory

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⊕

⊻

||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 |- ||direct sum |rowspan=3|The direct sum is a special way of combining several one modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).

|rowspan=3|Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = ∅) |- |align=center|direct sum of |- |align=right|Abstract algebra

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∀

||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

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∃

||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

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∃!

||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

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:=

≡

:⇔

||definition | rowspan=3|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. | 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

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≅

||congruence | rowspan=3|△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. | rowspan=3| |- |align=center|is congruent to |- |align=right|geometry

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{ , }

||set brackets | rowspan=3|{a,b,c} means the set consisting of a, b, and c. | rowspan=3|ℕ = { 1, 2, 3, …} |- |align=center|the set of … |- |align=right|set theory

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{ : }

{ | }

||set builder notation | rowspan=3|{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)}. | rowspan=3|{n ∈ ℕ : n² < 20} = { 1, 2, 3, 4} |- |align=center|the set of … such that |- |align=right|set theory

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∅

{ }

||empty set

| rowspan=3|∅ means the set with no elements. { } means the same. | rowspan=3|{n ∈ ℕ : 1 < n² < 4} = ∅ |- |align=center| the empty set |- |align=right|set theory

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∈

\notin

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

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

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⊆

⊂

||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.

(Some writers use the symbol ⊂ as if it were the same as ⊆.) | rowspan=3|(A ∩ B) ⊆ A

ℕ ⊂ ℚ

ℚ ⊂ ℝ |- |align=center|is a subset of |- |align=right|set theory

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⊇

⊃

||superset | rowspan=3|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 ⊇.) | rowspan=3|(A ∪ B) ⊇ B

ℝ ⊃ ℚ |- |align=center|is a superset of |- |align=right|set theory

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∪

||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

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∩

||set-theoretic intersection | rowspan=3|A ∩ B means the set that contains all those elements that A and B have in common. | rowspan=3|{x ∈ ℝ : x² = 1} ∩ ℕ = {1} |- |align=center|intersected with; intersect |- |align=right|set theory

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\Delta

||symmetric difference | rowspan=3| $A\Delta B$ means the set of elements in exactly one of A or B. | rowspan=3|{1,5,6,8} $\Delta$ {2,5,8} = {1,2,6} |- |align=center|symmetric difference |- |align=right|set theory

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∖

||set-theoretic complement | rowspan=3|A ∖ B means the set that contains all those elements of A that are not in B. | rowspan=3|{1,2,3,4} ∖ {3,4,5,6} = {1,2} |- |align=center|minus; without |- |align=right|set theory

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( )

||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

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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

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o

||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

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ℕ

N

||natural numbers | rowspan=3|N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention. | rowspan=3|ℕ = {|a| : a ∈ ℤ, a ≠ 0} |- |align=center|N |- |align=right|numbers

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ℤ

Z

||integers

| rowspan=3|ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ⁺ means {1, 2, 3, ...} = ℕ. | rowspan=3|ℤ = {p, -p : p ∈ ℕ} ∪ {0} |- |align=center|Z |- |align=right|numbers

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ℚ

Q

||rational numbers

| rowspan=3|ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. | rowspan=3|3.14000... ∈ ℚ

π ∉ ℚ |- |align=center|Q |- |align=right|numbers

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ℝ

R

||real numbers

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ℂ

C

||complex numbers

| rowspan=3|ℂ means {a + b i : a,b ∈ ℝ}. | rowspan=3|i = √(−1) ∈ ℂ |- |align=center|C |- |align=right|numbers |- ||arbitrary constant | rowspan=3| C can be any number, most likely unknown; usually occurs when calculating antiderivatives. | rowspan=3|if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C |- |align=center|C |- |align=right|integral calculus

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∞

||infinity | rowspan=3|∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. | rowspan=3|lim_x→0 1/|x| = ∞ |- |align=center|infinity |- |align=right|numbers

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π

||pi

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||…||

||norm | rowspan=3| || x || is the norm of the element x of a normed vector space. | rowspan=3| || x + y || ≤ || x || + || y || |- |align=center|norm of

length of |- |align=right| linear algebra

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∑

||summation | rowspan=3| $\sum _{k=1}^{n}{a_{k}}$ means a₁ + a₂ + … + a_n. | rowspan=3| $\sum _{k=1}^{4}{k^{2}}$ = 1² + 2² + 3² + 4²

= 1 + 4 + 9 + 16 = 30

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∏

||product | rowspan=3| $\prod _{k=1}^{n}a_{k}$ means a₁a₂···a_n. | rowspan=3| $\prod _{k=1}^{4}(k+2)$ = (1+2)(2+2)(3+2)(4+2)

= 3 × 4 × 5 × 6 = 360

|- ||Cartesian product | rowspan=3| $\prod _{i=0}^{n}{Y_{i}}$ means the set of all (n+1)-tuples

(y₀, …, y_n).

| rowspan=3| $\prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}$ |- |align=center|the Cartesian product of; the direct product of |- |align=right|set theory

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∐

||coproduct | rowspan=3| | rowspan=3| |- |align=center|coproduct over … from … to … of |- |align=right|category theory

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′

||derivative | rowspan=3|f ′(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. | rowspan=3|If f(x) := x², then f ′(x) = 2x |- |align=center|… prime

derivative of |- |align=right|calculus

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∫

||indefinite integral or antiderivative | rowspan=3|∫ f(x) dx means a function whose derivative is f. | rowspan=3| ∫x² dx = x³/3 + C |- |align=center|indefinite integral of

the antiderivative of |- |align=right|calculus

|- ||definite integral | rowspan=3|∫_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. | rowspan=3|∫₀^b x² dx = b³/3; |- |align=center|integral from … to … of … with respect to |- |align=right|calculus

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∇

||gradient | rowspan=3|∇f (x₁, …, x_n) is the vector of partial derivatives (∂f / ∂x₁, …, ∂f / ∂x_n). | rowspan=3|If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z) |- |align=center|del, nabla, gradient of |- |align=right|calculus

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∂

||partial derivative | rowspan=3| With f (x₁, …, x_n), ∂f/∂x_i is the derivative of f with respect to x_i, with all other variables kept constant. | rowspan=3| If f(x,y) := x²y, then ∂f/∂x = 2xy |- |align=center|partial derivative of |- |align=right|calculus

|- |boundary | rowspan=3| ∂M means the boundary of M | rowspan=3| ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2} |- |align=center|boundary of |- |align=right|topology

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⊥

||perpendicular | rowspan=3|x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y. | rowspan=3|If l ⊥ m and m ⊥ n then l || n. |- |align=center|is perpendicular to |- |align=right|geometry

|- ||bottom element | rowspan=3|x = ⊥ means x is the smallest element. | rowspan=3|∀x : x ∧ ⊥ = ⊥ |- |align=center|the bottom element |- |align=right|lattice theory

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||

||parallel | rowspan=3|x || y means x is parallel to y. | rowspan=3|If l || m and m ⊥ n then l ⊥ n. |- |align=center|is parallel to |- |align=right|geometry

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⊧

||entailment | rowspan=3| A ⊧ B means the sentence A entails the sentence B, that is every model in which A is true, B is also true. | rowspan=3| A ⊧ A ∨ ¬A |- |align=center|entails |- |align=right| model theory

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⊢

||inference | rowspan=3|x ⊢ y means y is derived from x. | rowspan=3| A → B ⊢ ¬B → ¬A |- |align=center|infers or is derived from |- |align=right|propositional logic, predicate logic

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◅

||normal subgroup | rowspan=3| N ◅ G means that N is a normal subgroup of group G. | rowspan=3| Z(G) ◅ G |- |align=center|is a normal subgroup of |- |align=right|group theory

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/

||quotient group | rowspan=3| G/H means the quotient of group G modulo its subgroup H. | rowspan=3| {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}} |- |align=center| mod |- |align=right| group theory

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≈

||isomorphism | rowspan=3| G ≈ H means that group G is isomorphic to group H | rowspan=3| Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group. |- |align=center | is isomorphic to |- |align=right| group theory |- |approximately equal | rowspan=3|x ≈ y means x is approximately equal to y | rowspan=3|π ≈ 3.14159 |- |align=center|is approximately equal to |- |align=right|everywhere |-

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~

||same order of magnitude | rowspan=3| m ~ n, means the quantities m and n have the general size.

(Note that ~ is used for an approximation that is poor, otherwise use ≈ .) | rowspan=3|2 ~ 5

8 × 9 ~ 100

but π² ≈ 10 |- |align=right|roughly similar

poorly approximates |- |align=right|Approximation theory

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<,>

||inner product | rowspan=3|<x,y> means the inner product between x and y, as defined in an inner product space. | rowspan=3|The standard inner product between two vectors x = (2, 3) and y = (-1, 5) is:
<x, y> = 2×-1 + 3×5 = 13 |- |align=center|inner product of |- |align=right|vector algebra

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⊗

||tensor product | rowspan=3| V ⊗ U means the tensor product of V and U. | rowspan=3| {1, 2, 3, 4} ⊗ {1,1,2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} |- |align=center| tensor product of |- |align=right| linear algebra |}

External links

Equality, inequality

Basic operators

Vector algebra

Set theory

See also

External links