Table of mathematical symbols - Wikipedia, the free encyclopedia Página 1 de 13 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 http://en.wikipedia.org/wiki/Table_of_mathematical_symbols 16/05/2007 Table of mathematical symbols - Wikipedia, the free encyclopedia Página 2 de 13 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 http://en.wikipedia.org/wiki/Table_of_mathematical_symbols 16/05/2007 Table of mathematical symbols - Wikipedia, the free encyclopedia Página 3 de 13 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), http://en.wikipedia.org/wiki/Table_of_mathematical_symbols 16/05/2007 Table of mathematical symbols - Wikipedia, the free encyclopedia Página 4 de 13 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. http://en.wikipedia.org/wiki/Table_of_mathematical_symbols 16/05/2007 Table of mathematical symbols - Wikipedia, the free encyclopedia Página 5 de 13 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 http://en.wikipedia.org/wiki/Table_of_mathematical_symbols 16/05/2007 Table of mathematical symbols - Wikipedia, the free encyclopedia Página 6 de 13 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.