GetWiki:Symbols Table

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 third column contains an informal definition, and the fourth column gives a short example.

Be aware that, in some cases, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.

Note: The large table below has a mysterious origin, but it has often served as a complex testbed for GetWiki’s new table rendering code.

Basic Mathematical Symbols

Symbol	Name	Explanation	Example
	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.	1 ? 2
	is not equal to; does not equal
	everywhere
+	addition	4 + 6 means the sum of 4 and 6.	2 + 7 = 9
	plus
	arithmetic
?	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
	arithmetic
	set theoretic complement	A ? B means the set that contains all the elements of A that are not in B	{1,2,3,4} ? {3,4,5,6}  =  {1,2}
	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
÷ /	division	6 ÷ 3 or 6/3 means the division of 6 by 3.	2 ÷ 4 = .5 12/4 = 3
	divided by
	arithmetic
? ? ?	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
? ?	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	¬(¬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	n < 4  ?  n >2  ?  n = 3 when n is a natural number
	and
	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	n ? 4  ?  n ? 2  ? n ? 3 when n is a natural number
	or
	propositional_logic, lattice_theory
? ?	exclusive or	A &o(lus; B is true when either A or B are true, but not when both are true	(¬A) &o(lus; A is always true, A &o(lus; A is always false
	xor
	propositional_logic, boolean 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 + 5 = 2n
	there exists
	predicate logic
:= ? :?	definition	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)
	is defined as
	everywhere
{ , }	set brackets	{a,b,c} means the set consisting of a, b, and c	N = {0,1,2,...}
	the set of ...
	set_theory
{ : } { | }	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}
	the set of ... such that ...
	naive set
? {}	empty set	{} means the set with no elements; ? is the same thing	{n ? N : 1 < n2 < 4} = {}
	empty set
	set_theory
? ?	set membership	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
	is an element of; is not an element of
	everywhere, set_theory
? ?	subset	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
	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	A ? B ? B; R ? Q
	is a superset of
	set_theory
?	set theoretic union	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
	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
	set theoretic complement	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}
	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
	everywhere
f:X?Y	function arrow	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) = x2
	from ... to
	functions
N ?	natural numbers	N means {0,1,2,3,...}, but see the article on natural numbers for a different convention.	{	a> : a ? Z} = N
	N
	numbers
Z ?	integers	Z means {...,?3,?2,?1,0,1,2,3,...}	{a :	a> ? N} = Z
	Z
	numbers
Q ?	rational numbers	Q means {p/q : p,q ? Z, q ? 0}	3.14 ? Q; ? ? Q
	Q
	numbers
R ?	real numbers	R means {limn?? an : ? n ? N: an ? Q, the limit exists}	? ? R; ?(?1) ? R
	R
	numbers
C ?	complex numbers	C means {a + bi : a,b ? R}	i = ?(?1) ? C
	C
	numbers
< >	strict_inequality	x < y means x is less than y; x > y means x is greater than y	x < y  ?  y > x
	is less than, is greater than
	partial orders
? ?	inequality	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
	is less than or equal to, is greater than or equal to
	partial orders
?	square root	?x means the positive number whose square is x	?(x2) =	x>
	the principal square root of; square root
	real numbers
?	infinity	? is an element of the extended_number_line that is greater than all real numbers; it often occurs in limits	limx?0 1/	x> = ?
	infinity
	numbers
?	pi	? means the ratio of a circle’s circumference to its diameter	A = ?r² is the area of a circle with radius r
	pi
	Euclidean geometry
!	factorial	n! is the product 1×2×...×n	4! = 24
	factorial
	combinatorics
		absolute value		x> means the distance in the real line (or the complex plane) between x and zero		a + bi> = ?(a2 + b2)
	absolute value of
	numbers
	Â		norm		x	is the norm of the element x of a normed vector space		x+y	?	x	+	y
	norm of; length of
	functional analysis
?	summation	?k=1n ak means a1 + a2 + ... + an	?k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30
	sum over ... from ... to ... of
	arithmetic
?	product	?k=1n ak means a1a2···an	?k=14 (k + 2) = (1  + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360
	product over ... from ... to ... of
	arithmetic
	cartesian product	?i=0nYi means the set of all (n+1)-tuples (y0,...,yn).	?n=13R = Rn
	the cartesian product of; the direct product of
	set_theory
?	integration	?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; ?x2 dx = x3/3
	integral from ... to ... of ... with respect to
	calculus
f ’	derivative	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
	derivative of f; f prime
	calculus
?	gradient	?f (x1, …, xn) is the vector of partial derivatives (df / dx1, …, df / dxn)	If f (x,y,z) = 3xy + z² then ?f = (3y, 3x, 2z)
	del, nabla, gradient of
	calculus
?	partial	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 derivative of
	calculus
?	perpendicular	x ? y means x is perpendicular to y; or more generally x is orthogonal to y.
	is perpendicular to
	orthogonality
	bottom element	x = ? means x is the smallest element.
	the bottom element
	lattice_theory
?	entailment	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.
	entails
	propositional_logic, predicate logic
?	inference	x vdash y means y is derived from x.
	infers or is derived from
	propositional_logic, predicate logic

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NOTE: If some of these symbols are used in an article intended for beginners, it may be a good idea to include a statement like the below, included with the {{msg:symbols}} code. This will help the article reach a broader audience:

This article uses mathematical symbols.

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External Links

• Official Code Chart
• Jeff Miller: ‘’Earliest Uses of Various Mathematical Symbols”
• TCAEP - Institute of Physics

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Some content adapted from the Wikinfo article “Table of mathematical symbols” under the GNU Free Documentation License.