Table of mathematical symbols

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 second line contains an informal definition, and the third line gives a short example.

Note: If some of the symbols don't display properly for you, then your browser does not completely implement the HTML 4 character entities, or you have to install additional fonts. You can check your browser here (http://www.alanwood.net/demos/ent4_frame.html).

Symbol	Name	reads as	Category
+	addition	plus	arithmetic
	4 + 6 = 10 means if 4 is added to 6, the sum, or result, is 10.
	43 + 65 = 108; 2 + 7 = 9
−	subtraction	minus	arithmetic
	9 − 4 = 5 means if 4 is subtracted from 9, the result will be 5.
	87 − 36 = 51
	negative sign	negative	arithmetic
	−3 means the number 3 less than 0.
	−(− 5) = 5
	set theoretic complement	minus; without	set theory
	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}
×	multiplication	times	arithmetic
	3 × 4 = 12 means if 4 is multiplied by 3, the result will be 12.
	7 × 8 = 56
	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)}
÷ /	division	divided by	arithmetic
	6 ÷ 3 = 2 or 6/3 = 2 means if 6 is divided by 3, the result is 2.
	2 ÷ 4 = .5; 12/4 = 3
⇒ →	material implication	implies; if .. then	propositional logic
	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 mentioned further down
	x = 2 ⇒ x² = 4 is true, but x² = 4 ⇒ x = 2 is in general false (since x could be −2)
⇔ ↔	material equivalence	if and only if; iff	propositional logic
	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
∧	logical conjunction or meet in a lattice	and	propositional logic, lattice theory
	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
∨	logical disjunction or join in a lattice	or	propositional logic, lattice theory
	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
$\oplus$ ⊻	exclusive or	xor	propositional logic, boolean algebra
$\oplus$ ⊻	$A \oplus B$ is true when either A or B are true, but not when both are true
¬	logical negation	not	propositional logic
	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 ∧ B) ⇔ (¬A) ∨ (¬B); `x` ∉ `S` ⇔ ¬(`x` ∈ `S`)
∀	universal quantification	for all; for any; for each	predicate logic
	∀ x: P(x) means P(x) is true for all x
	∀ n ∈ N: n² ≥ n
∃	existential quantification	there exists	predicate logic
	∃ x: P(x) means there is at least one x such that P(x) is true
	∃ n ∈ N: n + 5 = 2n
=	equality	equals	everywhere
	`x` = `y` means `x` and `y` are different names for precisely the same thing
	1 + 2 = 6 − 3
≠	Inequation	does not equal	everywhere
≠	x ≠ y States that x and y do not represent the same value.
:= ≡ :⇔	definition	is defined as	everywhere
	`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)
{ , }	set brackets	the set of ...	set theory
	{`a`,`b`,`c`} means the set consisting of `a`, `b`, and `c`
	N = {0,1,2,...}
{ : } { \| }	set builder notation	the set of ... such that ...	set theory
	{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} = {0,1,2,3,4}
∅ {}	empty set	empty set	set theory
	{} means the set with no elements; ∅ is the same thing
	{n ∈ N : 1 < n² < 4} = {}
∈ ∉	set membership	in; is in; is an element of; is a member of; belongs to	set theory
	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
⊆ ⊂	subset	is a subset of	set theory
	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
∪	set theoretic union	the union of ... and ...; union	set theory
	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
∩	set theoretic intersection	intersected with; intersect	set theory
	A ∩ B means the set that contains all those elements that A and B have in common
	{x ∈ R : x² = 1} ∩ N = {1}
\	set theoretic complement	minus; without	set theory
	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}
( )	function application	of	set theory
	`f`(`x`) means the value of the function `f` at the element `x`
	If `f`(`x`) := `x`², then `f`(3) = 3² = 9
	precedence grouping		everywhere
	perform the operations inside the parentheses first
	(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4
f:X→Y	function arrow	from ... to	functions
	`f`: `X` → `Y` means the function `f` maps the set `X` into the set `Y`
	Consider the function `f`: Z → N defined by `f`(`x`) = `x`²
N ℕ	natural numbers	N	numbers
	N means {0,1,2,3,...}, but see the article on natural numbers for a different convention.
	{\|a\| : a ∈ Z} = N
Z ℤ	integers	Z	numbers
	Z means {...,−3,−2,−1,0,1,2,3,...}
	{a : \|a\| ∈ N} = Z
Q ℚ	rational numbers	Q	numbers
	Q means {p/q : p,q ∈ Z, q ≠ 0}
	3.14 ∈ Q; π ∉ Q
R ℝ	real numbers	R	numbers
	R means {lim_n→∞ a_n : ∀ n ∈ N: `a`_n ∈ Q, the limit exists}
	π ∈ R; √(−1) ∉ R
C ℂ	complex numbers	C	numbers
	C means {a + bi : a,b ∈ R}
	i = √(−1) ∈ C
< >	strict inequality	is less than, is greater than	partial orders
	x < y means x is less than y; x > y means x is greater than y
	x < y ⇔ `y` > `x`
≤ ≥	inequality	is less than or equal to, is greater than or equal to	partial orders
	`x` ≤ `y` means `x` is less than or equal to `y`; x ≥ y means x is greater than or equal to y
	x ≥ 1 ⇒ x² ≥ x
√	square root	the principal square root of; square root	real numbers
	√x means the positive number whose square is x
	√(x²) = \|x\|
∞	infinity	infinity	numbers
	∞ 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\| = ∞
π	pi	pi	Euclidean geometry
	π means the ratio of a circle's circumference to its diameter
	A = πr² is the area of a circle with radius r
!	factorial	factorial	combinatorics
	n! is the product 1×2×...×n
	4! = 24
\| \|	absolute value	absolute value of	numbers
	\|`x`\| means the distance in the real line (or the complex plane) between `x` and zero
	\|a + bi\| = √(a² + b²)
\|\| \|\|	norm	norm of; length of	functional analysis
	\|\|`x`\|\| is the norm of the element x of a normed vector space
	\|\|x+y\|\| ≤ \|\|x\|\| + \|\|y\|\|
∑	summation	sum over ... from ... to ... of	arithmetic
	∑_k=1ⁿ a_k means a₁ + a₂ + ... + a_n
	∑_k=1⁴ k² = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
∏	product	product over ... from ... to ... of	arithmetic
	∏_k=1ⁿ a_k means a₁a₂···a_n
	∏_k=1⁴ (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360
	cartesian product	the cartesian product of; the direct product of	set theory
	∏_i=0ⁿY_i means the set of all (n+1)-tuples (y₀,...,y_n).
	∏_n=1³R = Rⁿ
∫	integration	integral from ... to ... of ... with respect to	calculus
	∫_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; ∫x² dx = x³/3
f '	derivative	derivative of f; f prime	calculus
	f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there
	If f(x) = x², then f '(x) = 2x and f ''(x) = 2
∇	gradient	del, nabla, gradient of	calculus
	∇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) A transparent image for text is: Image:Del.gif ().
∂	partial	partial derivative of	calculus
	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
⊥	perpendicular	is perpendicular to	orthogonality
	x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.
	.
	bottom element	the bottom element	lattice theory
	x = ⊥ means x is the smallest element.
	.
$\models$ ⊧	entailment	entails	propositional logic, predicate logic
$\models$ ⊧	$a \models b$ means the sentence a entails the sentence b. Formal definition: $a \models b$ if and only if, in every model in which a is true, b is also true.
$\vdash$ ⊢	inference	infers or is derived from	propositional logic, predicate logic
	x $\vdash$ y means y is derived from x.
	.
.	.
	insert more (suggestions are the inequality symbols); some symbols are used in examples before they are defined
	.

If some of these symbols are used in a AskFactMaster.Com 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:

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The article contains information about how to produce these math symbols in AskFactMaster.Com articles.

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

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Categories: Mathematical notation

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Table of mathematical symbols

+

−

×

÷/

⇒→

⇔↔

∧

∨

⊻

¬

∀

∃

=

≠

:=≡:⇔

{ , }

{ : }{ | }

∅{}

∈∉

⊆⊂

∪

∩

\

( )

f:X→Y

N

ℕ

Z

ℤ

Q

ℚ

R

ℝ

C

ℂ

<>

≤≥

√

∞

π

!

| |

|| ||

∑

∏

∫

f '

∇

∂

⊥

⊧

⊢

See also:

External links

÷

/

⇒
→

⇔
↔

:=
≡
:⇔

{ : }

{ | }

∅
{}

∈
∉

⊆
⊂

<
>

≤
≥