Table of Mathematical Symbols = ≠ <> != < > ≪ ≫ ≤ <=

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Table of mathematical symbols

From Wikipedia, the free encyclopedia

For the HTML codes of mathematical symbols see mathematical HTML. Note: This article contains special characters.

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

Name Symbol Read as Explanation Examples Category equality x = y means x and y is equal to; represent the same 1 + 1 = 2 = equals thing or value. everywhere ≠ inequation x ≠ y means that x and y do not represent the same thing or value. is not equal to; 1 ≠ 2 does not equal (The symbols != and <> <> are primarily from computer science. They are avoided in != everywhere mathematical texts. ) < strict inequality x < y means x is less than y.

> is less than, is x > y means x is greater 3 < 4 greater than, is than y. 5 > 4. much less than, is much greater x ≪ y means x is much 0.003 ≪ 1000000 ≪ than less than y.

x ≫ y means x is much greater than y. ≫ order theory

inequality x ≤ y means x is less than or equal to y. ≤ x ≥ y means x is <= is less than or greater than or equal to

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equal to, is y. greater than or The symbols and equal to ( <= 3 ≤ 4 and 5 ≤ 5 ≥ >= are primarily from 5 ≥ 4 and 5 ≥ 5 computer science. They >= order theory are avoided in mathematical texts. ) proportionality is proportional y ∝ x means that y = kx if y = 2 x, then y ∝ x ∝ to; varies as for some constant k. everywhere addition 4 + 6 means the sum of plus 2 + 7 = 9 4 and 6. arithmetic disjoint union + A + A means the the disjoint 1 2 A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒ disjoint union of sets A union of ... 1 A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), and ... and A2. (4,2), (5,2), (7,2)} set theory subtraction 9 − 4 means the minus 8 − 3 = 5 subtraction of 4 from 9. arithmetic negative sign −3 means the negative negative ; minus −(−5) = 5 of the number 3. − arithmetic set-theoretic A − B means the set complement that contains all the {1,2,4} − {1,3,4} = {2} minus; without elements of A that are set theory not in B. multiplication 3 × 4 means the times multiplication of 3 by 7 × 8 = 56 arithmetic 4. Cartesian product X×Y means the set of the Cartesian all ordered pairs with product of ... the first element of each {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)} × and ...; the direct pair selected from X product of ... and the second element and ... selected from Y. set theory cross product u × v means the cross (1,2,5) × (3,4,−1) = cross product of vectors u (−22, 16, − 2) vector algebra and v

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multiplication 3 · 4 means the times multiplication of 3 by 7 · 8 = 56 4. · arithmetic dot product u · v means the dot dot product of vectors u (1,2,5) · (3,4,−1) = 6 vector algebra and v ÷ division 2 ÷ 4 = .5 6 ÷ 3 or 6 ⁄ 3 means the divided by division of 6 by 3. 12 ⁄ 4 = 3 ⁄ arithmetic plus-minus 6 ± 3 means both 6 + 3 The equation x = 5 ± √4, has two plus or minus and 6 - 3. solutions, x = 7 and x = 3. arithmetic ± plus-minus 10 ± 2 or eqivalently 10 If a = 100 ± 1 mm, then a is ≥ 99 mm plus or minus ± 20% means the range and ≤ 101 mm. measurement from 10 − 2 to 10 + 2. minus-plus 6 ± (3 5) means both ∓ minus or plus 6 + (3 - 5) and 6 - (3 + cos( x ± y) = cos( x) cos( y) sin( x) sin( y). arithmetic 5). square root the principal √x means the positive square root of; number whose square is √4 = 2 square root x. real numbers complex square root √ if z = r exp( iφ) is the complex represented in polar square root of … coordinates with -π < φ √(-1) = i

≤ π, then √z = √r exp square root (i φ/2). complex numbers absolute value or |3| = 3 modulus |x| means the distance along the real line (or |–5| = |5| absolute value across the complex (modulus) of plane) between x and | i | = 1 zero. |…| numbers | 3 + 4 i | = 5 Euclidean distance Euclidean distance |x – y| means the For x = (1,1), and y = (4,5),

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2 2 between; Euclidean distance |x – y| = √([1–4] + [1–5] ) = 5 Euclidean norm between x and y. of Geometry Determinant |A| means the determinant of determinant of the Matrix theory matrix A divides A single vertical bar is used to denote Since 15 = 3×5, it is true that 3|15 and divides | divisibility. 5|15. Number Theory a|b means a divides b. factorial n ! is the product 1 × factorial 4! = 1 × 2 × 3 × 4 = 24 ! 2× ... × n. combinatorics transpose transpose T Swap rows for columns Aij = ( A )ji T matrix operations probability X ~ D , means the distribution random variable X has X ~ N(0,1), the standard normal has distribution the probability distribution distribution D. ~ statistics Row equivalence A~B means that B can is row equivalent be generated by using a to series of elementary Matrix theory row operations on A A ⇒ B means if A is material true then B is also true; implication if A is false then ⇒ nothing is said about B.

implies; if … → may mean the same 2 2 then as ⇒, or it may have the x = 2 ⇒ x = 4 is true, but x = 4 ⇒ x = → meaning for functions 2 is in general false (since x could be −2). given below.

propositional ⊃ may mean the same logic, Heyting ⊃ as ⇒, or it may have the algebra meaning for superset given below. material equivalence A ⇔ B means A is true ⇔ if B is true and A is x + 5 = y +2 ⇔ x + 3 = y if and only if; iff false if B is false.

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propositional ↔ logic The statement ¬A is logical negation true if and only if A is false.

A slash placed through ¬ not another operator is the ¬(¬A) ⇔ A same as "¬" placed in front. x ≠ y ⇔ ¬(x = y) ˜ propositional (The symbol ~ has logic many other uses, so ¬ or the slash notation is preferred. ) logical The statement A ∧ B is conjunction or true if A and B are both meet in a lattice true; else it is false. n < 4 ∧ n >2 ⇔ n = 3 when n is a and; min ∧ For functions A(x) and natural number. propositional B(x), A(x) ∧ B(x) is logic, lattice used to mean min(A(x), theory B(x)). A B logical The statement ∨ is disjunction or true if A or B (or both) join in a lattice are true; if both are false, the statement is false. n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a or; max ∨ natural number. For functions A(x) and propositional B(x), A(x) ∨ B(x) is logic, lattice theory used to mean max(A (x), B(x)). exclusive or The statement A ⊕ B is xor true when either A or (¬A) ⊕ A is always true, A ⊕ A is B, but not both, are propositional always false. logic, Boolean true. A B means the ⊕ algebra same. The direct sum is a

direct sum special way of Most commonly, for vector spaces U, V, combining several one and W, the following consequence is modules into one direct sum of used: general module (the U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = symbol ⊕ is used, is ∅) Abstract algebra only for logic).

universal quantification ∀ n ∈ : n2 ≥ n. for all; for any; ∀ x: P(x) means P(x) is

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for each true for all x. ∀ predicate logic existential quantification ∃ x: P(x) means there is ∃ n ∈ : n is even. ∃ there exists at least one x such that P(x) is true. predicate logic uniqueness quantification ∃! x: P(x) means there ∃! there exists is exactly one x such ∃! n ∈ : n + 5 = 2 n. exactly one that P(x) is true. predicate logic := x := y or x ≡ y means x definition is defined to be another name for y

cosh x := (1/2)(exp x + exp (−x)) (Some writers use ≡ to ≡ is defined as mean congruence ). A xor B : ⇔ (A ∨ B) ∧ ¬(A ∧ B)

P : ⇔ Q means P is everywhere defined to be logically :⇔ equivalent to Q.

congruence △ABC △DEF means triangle ABC is is congruent to congruent to (has the same measurements as) geometry triangle DEF. congruence relation ... is congruent a ≡ b (mod n) means a 5 ≡ 11 (mod 3) ≡ to ... modulo ... − b is divisible by n modular arithmetic set brackets {a,b,c} means the set { , } the set of … consisting of a, b, and = { 1, 2, 3, …} set theory c. set builder { : } notation {x : P(x)} means the set of all x for which P(x) the set of … {n : n2 < 20} = { 1, 2, 3, 4} is true. { x | P(x)} is the ∈ such that same as { x : P(x)}. { | } set theory

empty set ∅ means the set with 2 ∅ no elements. { } means {n ∈ : 1 < n < 4} = ∅ the empty set the same.

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{ } set theory set membership ∈ a ∈ S means a is an is an element of; (1/2) −1 is not an element ∈ element of the set S; a ∉ of S means a is not an 2−1 ∉ everywhere, set element of S. ∉ theory (subset) A ⊆ B means subset every element of A is also element of B. ⊆ (A ∩ B) ⊆ A (proper subset) A ⊂ B is a subset of ⊂ means A ⊆ B but A ≠ B.

⊂ (Some writers use the ⊂ set theory symbol ⊂ as if it were the same as ⊆.) A ⊇ B means every superset element of B is also element of A. ⊇ A ⊃ B means A ⊇ B but (A ∪ B) ⊇ B is a superset of A ≠ B. ⊃

⊃ (Some writers use the set theory symbol ⊃ as if it were the same as ⊇.) (exclusive) A ∪ B set-theoretic means the set that union contains all the elements from A, or all the elements from B, but not both. "A or B, but not both." the union of … A ⊆ B ⇔ ( A ∪ B) = B (inclusive) ∪ and (inclusive) A ∪ B means union the set that contains all the elements from A, or all the elements from B, or all the elements from set theory both A and B. "A or B or both". set-theoretic intersection A ∩ B means the set {x : x2 = 1} = {1} that contains all those ∈ ∩ intersected with; elements that A and B

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intersect have in common. ∩ set theory symmetric difference A∆B means the set of ∆ symmetric elements in exactly one {1,5,6,8} ∆ {2,5,8} = {1,2,6} difference of A or B. set theory set-theoretic A B means the set complement that contains all those {1,2,3,4} {3,4,5,6} = {1,2} minus; without elements of A that are set theory not in B. function application f(x) means the value of the function f at the 2 2 of If f(x) := x , then f(3) = 3 = 9. element x. set theory ( ) precedence grouping Perform the operations inside the parentheses (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. parentheses first. everywhere function arrow f: X → Y means the from … to 2 f:X→Y function f maps the set Let f: → be defined by f(x) := x . set theory,type X into the set Y. theory function composition fog is the function, such if f(x) := 2 x, and g(x) := x + 3, then ( fog) o composed with that ( fog)( x) = f(g(x)). (x) = 2( x + 3). set theory natural numbers N means { 1, 2, 3, ...}, but see the article on N = {| a| : a ∈ , a ≠ 0} natural numbers for a different convention. N numbers integers means {..., −3, −2, −1, 0, 1, 2, 3, ...} and Z = { p, -p : p ∈ } ∪ {0} + means {1, 2, 3, ...} = . Z numbers rational numbers means { p/q : p ∈ , 3.14000... ∈ Q q ∈ }. π Q numbers

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real numbers π ∈ R means the set of real

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

real or complex numbers because K means the statement holds substituting K for K R and also for C. K and

linear algebra . infinity ∞ is an element of the extended number line lim 1/| x| = ∞ infinity that is greater than all x→0 ∞ real numbers; it often numbers occurs in limits. norm norm of || x || is the norm of the ||…|| element x of a normed || x + y || ≤ || x || + || y || length of vector space. linear algebra summation = 1 2 + 2 2 + 3 2 + 4 2 sum over … means a + a + ∑ from … to … of 1 2 … + a . arithmetic n = 1 + 4 + 9 + 16 = 30 product = (1+2)(2+2)(3+2)(4+2) product over … means from … to … of ∏ a a ··· a . arithmetic 1 2 n = 3 × 4 × 5 × 6 = 360 Cartesian product

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the Cartesian product of; the direct product of means the set of all (n+1)-tuples set theory (y0, …, yn). coproduct coproduct over … from … to … of category 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. … prime 2 The dot notation If f(x) := x , then f ′(x) = 2 x derivative of indicates a time • derivative. That is

calculus .

indefinite integral or antiderivative indefinite ∫ f(x) d x means a 2 3 integral of function whose ∫x d x = x /3 + C derivative is f. the antiderivative ∫ of calculus definite integral b ( ) d means the ∫a f x x integral from … signed area between the b x 2 d = 3/3; to … of … with x-axis and the graph of ∫0 x b respect to the function f between calculus x = a and x = b. gradient (x , …, x ) is the ∇f 1 n del, nabla, vector of partial If f ( x,y,z) := 3 xy + z², then ∇f = (3 y, 3 x, ∇ gradient of derivatives ( ∂f / ∂x1, …, 2z) f calculus ∂ / ∂xn).

partial With f (x 1, …, xn), differential f/ x is the derivative ∂ ∂ i If f(x,y) := x 2y, then f/ x = 2xy partial, d of f with respect to x i, ∂ ∂ ∂ with all other variables calculus kept constant. boundary

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boundary of ∂M means the boundary ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2} topology of M

perpendicular x ⊥ y means x is is perpendicular perpendicular to y; or If l ⊥ m and m ⊥ n then l || n. to more generally x is geometry orthogonal to y. ⊥ bottom element the bottom x = ⊥ means x is the ∀x : x ∧ ⊥ = ⊥ element smallest element. lattice theory parallel x || y means x is parallel is parallel to If l || m and m ⊥ n then l ⊥ n. || to y. geometry entailment A B means the sentence A entails the entails sentence B, that is A A ∨ ¬A every model in which A model theory is true, B is also true. inference infers or is derived from x y means y is A → B ¬ B → ¬A propositional derived from x. logic, predicate logic normal subgroup N G means that N is is a normal a normal subgroup of Z(G) G subgroup of group G. group theory quotient group G/H means the quotient {0, a, 2 a, b, b+a, b+2 a} / {0, b} = {{0, mod of group G modulo its b}, { a, b+a}, {2 a, b+2 a}} group theory subgroup H. / quotient set A/~ means the set of all If we define ~ by x~y ⇔ x-y∈Z, then mod ~ equivalence classes in R/~ = {{ x+n : n∈Z} : x ∈ (0,1]} set theory A. isomorphism G ≈ H means that Q / {1, −1} ≈ V, is isomorphic to group G is isomorphic where Q is the quaternion group and V is group theory to group H the Klein four-group. ≈ approximately equal x ≈ y means x is is approximately approximately equal to π ≈ 3.14159 equal to y everywhere

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same order of magnitude m ~ n, means the roughly similar quantities m and n have 2 ~ 5 the general size. poorly 8 × 9 ~ 100 ~ approximates (Note that ~ is used for an approximation that Approximation but π2 ≈ 10 theory is poor, otherwise use ≈ .)

〈,〉 inner product 〈x,y〉 means the ( | ) inner product of x and y The standard inner product between two as defined in an inner vectors x = (2, 3) and y = (−1, 5) is: product space. 〈x, y 〉 = 2×−1 + 3×5 = 13 inner product of < , > For spatial vectors, the A:B = A B dot product notation, ∑ ij ij x·y is common. i,j For matricies, the colon · notation may be used. vector algebra : tensor product V U means the tensor {1, 2, 3, 4} {1,1,2} = tensor product of product of V and U. {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} linear algebra convolution f * g means the * convolution convolution of f and g. mean is the mean (average overbar . value of xi). statistics

delta equal to means equal by definition. When is used, equality is not equal by true generally, but . definition rather equality is true under certain assumptions that are everywhere taken in context.