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 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.

Basic mathematical symbols

rowspan="3" align=center\| Symbol	align=left\|Name	rowspan="3" \|Explanation	rowspan="3" \|Example
lign=center\|Should be read as
lign=right\|Category
rowspan=3 bgcolor=#d0f0d0 align=center\| =	\|equality	rowspan=3\|`x` = `y` means `x` and `y` represent the same thing or value.	rowspan=3\|1 + 1 = 2
lign=center\|is equal to; equals
lign=right\|everywhere
rowspan=3 bgcolor=#d0f0d0 align=center\| ≠	\|Inequation	rowspan=3\| x ≠ y means that x and y do not represent the same thing or value.	rowspan=3\|1 ≠ 2
lign=center\|is not equal to; does not equal
lign=right\|everywhere
rowspan=3 bgcolor=#d0f0d0 align=center\| +	addition	rowspan=3\|4 + 6 means the sum of 4 and 6.	rowspan=3\|2 + 7 = 9
lign=center\|plus
lign=right\|arithmetic
rowspan=9 bgcolor=#d0f0d0 align=center\| −	subtraction	rowspan=3\|9 − 4 means the subtraction of 4 from 9.	rowspan=3\|8 − 3 = 5
lign=center\|minus
lign=right\|arithmetic
a href="/encyclopedia/negative-and-non-negative-numbers" title="negative and non-negative numbers">negative sign	rowspan=3\|−3 means the negative of the number 3.	rowspan=3\|−(−5) = 5
lign=center\|negative
lign=right\|arithmetic
a href="/encyclopedia/set-theoretic-complement" title="set theoretic complement">set theoretic complement	rowspan=3\|A − B means the set that contains all the elements of A that are not in B.	rowspan=3\|{1,2,4} − {1,3,4} = {2}
lign=center\|minus; without
lign=right\|set theory
rowspan=6 bgcolor=#d0f0d0 align=center\| ×	multiplication	rowspan=3\|3 × 4 means the multiplication of 3 by 4.	rowspan=3\|7 × 8 = 56
lign=center\|times
lign=right\|arithmetic
a href="/encyclopedia/Cartesian-product" title="Cartesian product">Cartesian product	rowspan=3\|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.	rowspan=3\|{1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
lign=center\|the Cartesian product of and ; the direct product of and
lign=right\|set theory
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
lign=center\|divided by
lign=right\|arithmetic
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).
lign=center\|implies; if .. then
lign=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
lign=center\|if and only if; iff
lign=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`)
lign=center\|not
lign=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.
lign=center\|and
lign=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.
lign=center\|or
lign=right\|propositional logic, lattice theory
rowspan=3 bgcolor=#d0f0d0 align=center\| ⊕ ⊻	exclusive or	rowspan=3\| $A \oplus B$ is true when either A or B is true, but not when both are true.	rowspan=3\| (¬A) $\oplus$ A is always true, A $\oplus$ A is always false.
lign=center\|xor
lign=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
lign=center\|for all; for any; for each
lign=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 + 5 = 2n
lign=center\|there exists
lign=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)
lign=center\|is defined as
lign=right\|everywhere
rowspan=3 bgcolor=#d0f0d0 align=center\| { , }	\|set brackets	rowspan=3\|{`a`,`b`,`c`} means the set consisting of `a`, `b`, and `c`.	rowspan=3\|N = {0,1,2,...}
lign=center\|the set of ...
lign=right\|set theory
rowspan=3 bgcolor=#d0f0d0 align=center\| { : } { \| }	\|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} = {0,1,2,3,4}
lign=center\|the set of ... such that ...
lign=right\|set theory
rowspan=3 bgcolor=#d0f0d0 align=center\| {}	empty set	rowspan=3\| means the set with no elements. {} means the same.	rowspan=3\|{n ∈ N : 1 < n² < 4} =
lign=center\| the empty set
lign=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
lign=center\|is an element of; is not an element of
lign=right\|everywhere, set theory
rowspan=3 bgcolor=#d0f0d0 align=center\| ⊆ ⊂	\|subset	rowspan=3\|A ⊆ B means every element of A is also element of B. A ⊂ B means `A` ⊆ `B` but A ≠ B.	rowspan=3\|A ∩ B ⊆ A; Q ⊂ R
lign=center\|is a subset of
lign=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
lign=center\|is a superset of
lign=right\|set theory
rowspan=3 bgcolor=#d0f0d0 align=center\| ∪	\|set theoretic union	rowspan=3\|A ∪ B means the set that contains all the elements from A and also all those from B, but no others.	rowspan=3\|A ⊆ B ⇔ A ∪ B = B
lign=center\|the union of ... and ...; union
lign=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\|{x ∈ R : x² = 1} ∩ N = {1}
lign=center\|intersected with; intersect
lign=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\|{1,2,3,4} \ {3,4,5,6} = {1,2}
lign=center\|minus; without
lign=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`) := `x`², then `f`(3) = 3² = 9.
lign=center\|of
lign=right\|set theory
precedence grouping	rowspan=3\| Perform the operations inside the parentheses first.	rowspan=3\|(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
lign=center\|
lign=right\|everywhere
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`) = `x`².
lign=center\|from ... to
lign=right\|functions
rowspan=3 bgcolor=#d0f0d0 align=center\| N ℕ	\|natural numbers	rowspan=3\|N means {0,1,2,3,...}, but see the article on natural numbers for a different convention.	rowspan=3\|{\|a\| : a ∈ Z} = N
lign=center\|N
lign=right\|numbers
rowspan=3 bgcolor=#d0f0d0 align=center\| Z ℤ	integers	rowspan=3\|Z means {...,−3,−2,−1,0,1,2,3,...}.	rowspan=3\|{a : \|a\| ∈ N} = Z
lign=center\|Z
lign=right\|numbers
rowspan=3 bgcolor=#d0f0d0 align=center\| Q ℚ	rational numbers	rowspan=3\|Q means {p/q : p,q ∈ Z, q ≠ 0}.	rowspan=3\|3.14 ∈ Q π ∉ Q
lign=center\|Q
lign=right\|numbers
rowspan=3 bgcolor=#d0f0d0 align=center\| R ℝ	real numbers	rowspan=3\|R means {lim_n→∞ a_n : ∀ n ∈ N: `a`_n ∈ Q, the limit exists}.	rowspan=3\|π ∈ R √(−1) ∉ R
lign=center\|R
lign=right\|numbers
rowspan=3 bgcolor=#d0f0d0 align=center\| C ℂ	complex numbers	rowspan=3\|C means {a + bi : a,b ∈ R}.	rowspan=3\|i = √(−1) ∈ C
lign=center\|C
lign=right\|numbers
rowspan=3 bgcolor=#d0f0d0 align=center\| < >	\|strict inequality	rowspan=3\|x < y means x is less than y. x > y means x is greater than y.	rowspan=3\|x < y ⇔ `y` > `x`
lign=center\|is less than, is greater than
lign=right\|partial orders
rowspan=3 bgcolor=#d0f0d0 align=center\| ≤ ≥	\|inequality	rowspan=3\|`x` ≤ `y` means `x` is less than or equal to `y`. x ≥ y means x is greater than or equal to y.	rowspan=3\|x ≥ 1 ⇒ x² ≥ x
lign=center\|is less than or equal to, is greater than or equal to
lign=right\|partial orders
rowspan=3 bgcolor=#d0f0d0 align=center\| √	\|square root	rowspan=3\|√x means the positive number whose square is x.	rowspan=3\|√(x²) = \|x\|
lign=center\|the principal square root of; square root
lign=right\|real numbers
rowspan=3 bgcolor=#d0f0d0 align=center\| ∞	\|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\| = ∞
lign=center\|infinity
lign=right\|numbers
rowspan=3 bgcolor=#d0f0d0 align=center\| π	pi	rowspan=3\|π means the ratio of a circle's circumference to its diameter.	rowspan=3\|A = πr² is the area of a circle with radius r
lign=center\|pi
lign=right\|Euclidean geometry
rowspan=3 bgcolor=#d0f0d0 align=center\| !	\|factorial	rowspan=3\|n! is the product 1×2×...×n.	rowspan=3\|4! = 1 × 2 × 3 × 4 = 24
lign=center\|factorial
lign=right\|combinatorics
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\| \|a + bi\| = √(a² + b²)
lign=center\|absolute value of
lign=right\|numbers
rowspan=3 bgcolor=#d0f0d0 align=center\| \|\| \|\|	\|norm	rowspan=3\| \|\|`x`\|\| is the norm of the element x of a normed vector space.	rowspan=3\| \|\|x+y\|\| ≤ \|\|x\|\| + \|\|y\|\|
lign=center\|norm of; length of
lign=right\|functional analysis
rowspan=3 bgcolor=#d0f0d0 align=center\| ∑	\|summation	rowspan=3\|∑_k=1ⁿ a_k means a₁ + a₂ + ... + a_n.	rowspan=3\|∑_k=1⁴ k² = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
lign=center\|sum over ... from ... to ... of
lign=right\|arithmetic
rowspan=6 bgcolor=#d0f0d0 align=center\| ∏	\|product	rowspan=3\|∏_k=1ⁿ a_k means a₁a₂···a_n.	rowspan=3\|∏_k=1⁴ (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360
lign=center\|product over ... from ... to ... of
lign=right\|arithmetic
Cartesian product	rowspan=3\|∏_i=0ⁿY_i means the set of all (n+1)-tuples (y₀,...,y_n).	rowspan=3\|∏_n=1³R = Rⁿ
lign=center\|the Cartesian product of; the direct product of
lign=right\|set theory
rowspan=3 bgcolor=#d0f0d0 align=center\| ∫	\|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; ∫x² dx = x³/3
lign=center\|integral from ... to ... of ... with respect to
lign=right\|calculus
rowspan=3 bgcolor=#d0f0d0 align=center\| f '	\|derivative	rowspan=3\|f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there.	rowspan=3\|If f(x) = x², then f '(x) = 2x and f ''(x) = 2
lign=center\|derivative of f; f prime
lign=right\|calculus
rowspan=3 bgcolor=#d0f0d0 align=center\| ∇	\|gradient	rowspan=3\|∇f (x₁, …, x_n) is the vector of partial derivatives (df / dx₁, …, df / dx_n).	rowspan=3\|If f (x,y,z) = 3xy + z² then ∇f = (3y, 3x, 2z) A transparent image for text is: Image:Del.gif ().
lign=center\|del, nabla, gradient of
lign=right\|calculus
rowspan=3 bgcolor=#d0f0d0 align=center\| ∂	\|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
lign=center\|partial derivative of
lign=right\|calculus
rowspan=6 bgcolor=#d0f0d0 align=center\| ⊥	\|perpendicular	rowspan=3\|x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.	rowspan=3\|
lign=center\|is perpendicular to
lign=right\|orthogonality
bottom element	rowspan=3\|x = ⊥ means x is the smallest element.	rowspan=3\|
lign=center\|the bottom element
lign=right\|lattice theory
rowspan=3 bgcolor=#d0f0d0 align=center\| ⊧	\|entailment	rowspan=3\| $a \models b$ 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.	rowspan=3\|
lign=center\|entails
lign=right\|propositional logic, predicate logic
rowspan=3 bgcolor=#d0f0d0 align=center\| ⊢	\|inference	rowspan=3\|x $\vdash$ y means y is derived from x.	rowspan=3\|
lign=center\|infers or is derived from
lign=right\|propositional logic, predicate logic

If some of these symbols are used in a Wikipedia 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:

''This article uses [[table of mathematical symbols|mathematical symbols]].''

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