What are the symbols of sets?
SymbolsSymbolin HTMLSymbolin TEX Name Explanation Examples Read as Category =equalityis equal to;equalseverywherex = y means x and y represent the same thing or value. 2 = 21 + 1 = 2 ≠inequalityis not equal to;does not equaleverywherex ≠ y means that x and y do not represent the same thing or value.(The forms !=, /= or are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 2 + 2 ≠ 5 strict inequalityis less than,is greater thanorder theoryx < y means x is less than y.x > y means x is greater than y. 3 < 45 > 4 proper subgroupis a proper subgroup ofgroup theoryH < G means H is a proper subgroup of G. 5Z < ZA3 < S3 ≪≫(very) strict inequalityis much less than,is much greater thanorder theoryx ≪ y means x is much less than y.x ≫ y means x is much greater than y. 0.003 ≪ 1000000 asymptotic comparisonis of smaller order than,is of greater order thananalytic number theoryf ≪ g means the growth of f is asymptotically bounded by g.(This is I. M. Vinogradov's notation. Another notation is the Big O notation, which looks like f = O(g).) x ≪ ex ≤≥inequalityis less than or equal to,is greater than or equal toorder theoryx ≤ y means x is less than or equal to y.x ≥ y means x is greater than or equal to y.(The forms = are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 3 ≤ 4 and 5 ≤ 55 ≥ 4 and 5 ≥ 5 subgroupis a subgroup ofgroup theoryH ≤ G means H is a subgroup of G. Z ≤ ZA3 ≤ S3 reductionis reducible tocomputational complexity theoryA ≤ B means the problemA can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction. Ifthen≺ Karp reductionis Karp reducible to;is polynomial-time many-one reducible tocomputational complexity theoryL1 ≺ L2 means that the problem L1 is Karp reducible to L2.[1]If L1 ≺ L2 and L2 ∈ , then L1 ∈ P. ∝proportionalityis proportional to;varies aseverywherey ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x. Karp reduction[2]is Karp reducible to;is polynomial-time many-one reducible tocomputational complexity theoryA ∝ B means the problemA can be polynomially reduced to the problem B. If L1 ∝ L2 and L2 ∈ , then L1 ∈ P. +additionplus;addarithmetic4 + 6 means the sum of 4 and 6. 2 + 7 = 9 disjoint unionthe disjoint union of ... and ...set theoryA1 + A2 means the disjoint union of sets A1 and A2. A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, 9, 10} ⇒A1 + A2 = {(3,1), (4,1), (5,1), (6,1), (7,2), (8,2), (9,2), (10,2)} −subtractionminus;take;subtractarithmetic9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5 negative signnegative;minus;the opposite ofarithmetic−3 means the negative of the number 3. −(−5) = 5 set-theoretic complementminus;withoutset theoryA − B means the set that contains all the elements of A that are not in B.(∖ can also be used for set-theoretic complement as described below.) {1,2,4} − {1,3,4} = {2} ±plus-minusplus or minusarithmetic6 ± 3 means both 6 + 3 and 6 − 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3. plus-minusplus or minusmeasurement10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm. ∓minus-plusminus or plusarithmetic6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5). cos(x± y) = cos(x) cos(y) ∓ sin(x) sin(y). ×multiplicationtimes;multiplied byarithmetic3 × 4 means the multiplication of 3 by 4.(The symbol * is generally used in programming languages, where ease of typing and use of ASCIItext is preferred.) 7 × 8 = 56 Cartesian productthe Cartesian product of ... and ...;the direct product of ... and ...set theoryX×Y means the set of all ordered pairswith 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)} cross productcrosslinear algebrau × v means the cross product of vectorsu and v (1,2,5) × (3,4,−1) =(−22, 16, − 2) group of unitsthe group of units ofring theoryR× consists of the set of units of the ring R, along with the operation of multiplication.This may also be written R* as described below, orU(R). *multiplicationtimes;multiplied byarithmetica * b means the product of a and b.(Multiplication can also be denoted with × or ⋅, or even simple juxtaposition. * is generally used where ease of typing and use of ASCII text is preferred, such as programming languages.) 4 * 3 means the product of 4 and 3, or 12. convolutionconvolution;convolved withfunctional analysisf * g means the convolution of f and g. . complex conjugateconjugatecomplex numbersz* means the complex conjugate of z.( can also be used for the conjugate of z, as described below.) . group of unitsthe group of units ofring theoryR* consists of the set of units of the ring R, along with the operation of multiplication.This may also be written R× as described above, orU(R). hyperreal numbersthe (set of) hyperrealsnon-standard analysis*R means the set of hyperreal numbers. Other sets can be used in place of R. *N is the hypernaturalnumbers. Hodge dualHodge dual;Hodge starlinear algebra*v means the Hodge dual of a vector v. If vis a k-vectorwithin an n-dimensionalorientedinner productspace, then *v is an (n−k)-vector. If are the standard basis vectors of , ·multiplicationtimes;multiplied byarithmetic3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56 dot productdotlinear algebrau · v means the dot product of vectorsu and v (1,2,5) · (3,4,−1) = 6 placeholder(silent)functional analysisA · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. ⊗tensor product, tensor product of modulestensor product oflinear algebrameans the tensor product of V and U.[3]means the tensor product of modules Vand U over the ringR. {1, 2, 3, 4} ⊗ {1, 1, 2} ={{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} ÷⁄division(Obelus)divided by;overarithmetic6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = 0.512 ⁄ 4 = 3 quotient groupmodgroup theoryG / H means the quotient of group Gmodulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a,b+2a}} quotient setmodset theoryA/~ means the set of all ~ equivalence classes in A. If we define ~ by x ~ y ⇔ x − y ∈ ℤ, thenℝ/~ = { {x + n : n ∈ ℤ } : x ∈ [0,1) } √square rootthe (principal) square root ofreal numbersmeans the nonnegative number whose square is . complex square rootthe (complex) square root ofcomplex numbersif is represented in polar coordinates with , then . xmeanoverbar;… barstatistics(often read as "x bar") is the mean (average value of ). . complex conjugateconjugatecomplex numbersmeans the complex conjugate of z.(z* can also be used for the conjugate of z, as described above.) . finite sequence, tuplefinite sequence, tuplemodel theorymeans the finite sequence/tuple . . algebraic closurealgebraic closure offield theoryis the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as because it is the algebraic closure of the rational numbers . topological closure(topological) closure oftopologyis the topological closure of the set S.This may also be denoted as cl(S) orCl(S). In the space of the real numbers, (the rational numbers are dense in the real numbers). |…|absolute value;modulusabsolute value of; modulus ofnumbers|x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3|-5| = |5| = 5| i | = 1| 3 + 4i | = 5 Euclidean norm or Euclidean length or magnitudeEuclidean norm ofgeometry|x| means the (Euclidean) length of vectorx. For x = (3,-4)determinantdeterminant ofmatrix theory|A| means the determinant of the matrix A cardinalitycardinality of;size of;order ofset theory|X| means the cardinality of the set X.(# may be used instead as described below.) |{3, 5, 7, 9}| = 4. …normnorm of;length oflinear algebrax means the norm of the element x of a normed vector space.[4]x + y ≤ x + y nearest integer functionnearest integer tonumbersx means the nearest integer to x.(This may also be written [x], ⌊x⌉, nint(x) orRound(x).) 1 = 1, 1.6 = 2, −2.4 = −2, 3.49 = 3 ∣∤divisor, dividesdividesnumber theorya|b means a divides b.a∤b means a does not divide b.(This symbol can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar |character can be used.) Since 15 = 3×5, it is true that 3|15 and 5|15. conditional probabilitygivenprobabilityP(A|B) means the probability of the event a occurring given that b occurs. if X is a uniformly random day of the year P(X is May 25 | X is in May) = 1/31 restrictionrestriction of … to …;restricted toset theoryf|A means the function f restricted to the set A, that is, it is the function with domainA ∩ dom(f) that agrees with f. The function f : R → R defined by f(x) = x2 is not injective, but f|R+ is injective. such thatsuch that;so thateverywhere| means "such that", see ":" (described below). S = {(x,y) | 0 < y < f(x)}The set of (x,y) such that y is greater than 0 and less than f(x).parallelis parallel togeometryx y means x is parallel to y. If l m and m ⊥ n then l ⊥ n. incomparabilityis incomparable toorder theoryx y means x is incomparable to y. {1,2} {2,3} under set containment. exact divisibilityexactly dividesnumber theorypa n means pa exactly divides n (i.e. pa divides nbut pa+1 does not). 23 360. #cardinalitycardinality of;size of;order ofset theory#X means the cardinality of the set X.(|…| may be used instead as described above.) #{4, 6, 8} = 3 connected sumconnected sum of;knot sum of;knot composition oftopology, knot theoryA#B is the connected sum of the manifolds Aand B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition. A#Sm is homeomorphicto A, for any manifold A, and the sphere Sm. primorialprimorialnumber theoryn# is product of all prime numbers less than or equal to n. 12# = 2 × 3 × 5 × 7 × 11 = 2310 ℵaleph numberalephset theoryℵα represents an infinite cardinality (specifically, theα-th one, where α is an ordinal). |ℕ| = ℵ0, which is called aleph-null. ℶbeth numberbethset theoryℶα represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ). ?cardinality of the continuumcardinality of the continuum;c;cardinality of the real numbersset theoryThe cardinality of is denoted by or by the symbol (a lowercase Frakturletter C). :such thatsuch that;so thateverywhere: means "such that", and is used in proofs and theset-builder notation (described below). ∃ n ∈ ℕ: n is even. field extensionextends;overfield theoryK : F means the field K extends the field F.This may also be written as K ≥ F. ℝ : ℚ inner productof matricesinner product oflinear algebraA : B means the Frobenius inner product of the matrices A and B.The general inner product is denoted by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as described below. For spatial vectors, the dot product notation, x·y is common.See also Bra-ket notation. index of a subgroupindex of subgroupgroup theoryThe index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G !factorialfactorialcombinatoricsn! means the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24 logical negationnotpropositional logicThe statement !A is true if and only if A is false.A slash placed through another operator is the same as "!" placed in front.(The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation¬Ais preferred.) !(!A) ⇔ Ax ≠ y ⇔ !(x = y) ~probability distributionhas distributionstatisticsX ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution row equivalenceis row equivalent tomatrix theoryA~B means that B can be generated by using a series of elementary row operations on A same order of magnituderoughly similar;poorly approximatesapproximation theorym ~ n means the quantities m and nhave the sameorder of magnitude, or general size.(Note that ~ is used for an approximation that is poor, otherwise use ≈ .) 2 ~ 58 × 9 ~ 100but π2 ≈ 10 asymptotically equivalentis asymptotically equivalent toasymptotic analysisf ~ g means . x ~ x+1 equivalence relationare in the same equivalence classeverywherea ~ b means (and equivalently ). 1 ~ 5 mod 4 ≈approximately equalis approximately equal toeverywherex ≈ y means x is approximately equal to y.This may also be written ≃, ≅, ~, ♎ (Libra Symbol),or≒. π ≈ 3.14159 isomorphismis isomorphic togroup theoryG ≈ H means that group G is isomorphic (structurally identical) to group H.(≅ can also be used for isomorphic, as described below.) Q / {1, −1} ≈ V,where Q is the quaternion group and V is the Klein four-group. ≀wreath productwreath product of … by …group theoryA ≀ H means the wreath product of the group A by the group H.This may also be written A wr H. is isomorphic to the automorphismgroup of thecomplete bipartite graph on (n,n) vertices. ◅▻normal subgroupis a normal subgroup ofgroup theoryN ◅ G means that N is a normal subgroup of groupG. Z(G) ◅ G idealis an ideal ofring theoryI ◅ R means that I is an ideal of ring R. (2) ◅ Z antijointhe antijoin ofrelational algebraR ▻ S means the antijoin of the relations Rand S, the tuples in R for which there is not a tuple in S that is equal on their common attribute names. RS = R - R S ⋉⋊semidirect productthe semidirect product ofgroup theoryN ⋊φ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ. Also, if G = N ⋊φ H, then G is said to split over N.(⋊ may also be written the other way round, as ⋉, or as ×.) semijointhe semijoin ofrelational algebraR ⋉ S is the semijoin of the relations Rand S, the set of all tuples in R for which there is a tuple in Sthat is equal on their common attribute names. R S = a1,..,an(R S) ⋈natural jointhe natural join ofrelational algebraR ⋈ S is the natural join of the relations R and S, the set of all combinations of tuples in R and S that are equal on their common attribute names. ∴thereforetherefore;so;henceeverywhereSometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal. ∵becausebecause;sinceeverywhereSometimes used in proofs before reasoning. 3331 is prime ∵ it has no positive integer factors other than itself and one. ■□∎▮‣end of proofQED;tombstone;Halmos symboleverywhereUsed to mark the end of a proof.(May also be written Q.E.D.) D'Alembertiannon-Euclidean Laplacianvector calculusIt is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions. ⇒→⊃material implicationimplies;if … thenpropositional logic, Heyting algebraA ⇒ 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 functionsgiven below.)(⊃ may mean the same as ⇒,[5]or it may have the meaning for supersetgiven below.) x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since xcould be −2). ⇔↔material equivalenceif and only if;iffpropositional logicA ⇔ B means A is true if B is true and A is false if Bis false. x + 5 = y+ 2 ⇔ x + 3 = y ¬˜logical negationnotpropositional logicThe statement ¬A is true if and only if A is false.A slash placed through another operator is the same as "¬" placed in front.(The symbol ~ has many other uses, so ¬ or the slash notation is preferred. Computer scientists will often use! but this is avoided in mathematical texts.) ¬(¬A) ⇔ Ax ≠ y ⇔ ¬(x = y) ∧logical conjunction or meetin a latticeand;min;meetpropositional logic, lattice theoryThe statement A ∧ B is true if A and B are both true; else it is false.For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)). n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number. wedge productwedge product;exterior productexterior algebrau ∧ v means the wedge product of any multivectorsuand v. In three dimensional Euclidean space the wedge product and the cross product of two vectorsare each other's Hodge dual. exponentiation… (raised) to the power of …everywherea ^ b means a raised to the power of b(a ^ b is more commonly writtenab. The symbol ^ is generally used in programming languages where ease of typing and use of plain ASCII text is preferred.) 2^3 = 23 = 8 ∨logical disjunction or joinin a latticeor;max;joinpropositional logic, lattice theoryThe statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)). n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. ⊕⊻exclusive orxorpropositional logic, Boolean algebraThe statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. (¬A) ⊕ A is always true, A ⊕ A is always false. direct sumdirect sum ofabstract algebraThe direct sum is a special way of combining several objects into one general object.(The bun symbol ⊕, or the coproductsymbol ∐, is used; ⊻ is only for logic.) Most commonly, for vector spaces U, V, and W, the following consequence is used:U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0}) ∀universal quantificationfor all;for any;for eachpredicate logic∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n. ∃existential quantificationthere exists;there is;there arepredicate logic∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even. ∃!uniqueness quantificationthere exists exactly onepredicate logic∃! x: P(x) means there is exactly one x such thatP(x) is true. ∃! n ∈ ℕ: n + 5 = 2n. =::=≡:⇔≜≝≐definitionis defined as;is equal by definition toeverywherex := y, y =: x or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.(Some writers use ≡ to mean congruence).P :⇔ Q means P is defined to be logically equivalentto Q. ≅congruenceis congruent togeometry△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. isomorphicis isomorphic toabstract algebraG ≅ H means that group G is isomorphic (structurally identical) to group H.(≈ can also be used for isomorphic, as described above.) . ≡congruence relation... is congruent to ... modulo ...modular arithmetica ≡ b (mod n) means a − b is divisible by n 5 ≡ 2 (mod 3) { , }setbracketsthe set of …set theory{a,b,c} means the set consisting of a, b, and c.[6]ℕ = { 1, 2, 3, …} { : }{ | }{ ; }set builder notationthe set of … such thatset theory{x : P(x)} means the set of all xfor which P(x) is true.[6]{x | P(x)} is the same as {x : P(x)}. {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4} ∅{ }empty setthe empty setset theory∅ means the set with no elements.[6]{ } means the same. {n ∈ ℕ : 1 < n2 < 4} = ∅ ∈∉set membershipis an element of;is not an element ofeverywhere, set theorya ∈ S means a is an element of the set S;[6]a ∉ Smeans a is not an element of S.[6](1/2)−1 ∈ ℕ2−1 ∉ ℕ ⊆⊂subsetis a subset ofset theory(subset) A ⊆ B means every element of A is also an element of B.[7](proper subset) A ⊂ B means A ⊆ B but A ≠ B.(Some writers use the symbol ⊂ as if it were the same as ⊆.) (A ∩ B) ⊆ Aℕ ⊂ ℚℚ ⊂ ℝ ⊇⊃supersetis a superset ofset theoryA ⊇ B means every element of B is also an element of A.A ⊃ B means A ⊇ B but A ≠ B.(Some writers use the symbol ⊃ as if it were the same as ⊇.) (A ∪ B) ⊇ Bℝ ⊃ ℚ ∪set-theoretic unionthe union of … or …;unionset theoryA ∪ B means the set of those elements which are either in A, or in B, or in both.[7]A ⊆ B ⇔ (A ∪ B) = B ∩set-theoretic intersectionintersected with;intersectset theoryA ∩ B means the set that contains all those elements that A and B have in common.[7]{x ∈ ℝ : x2 = 1} ∩ ℕ = {1} ∆symmetric differencesymmetric differenceset theoryA ∆ B means the set of elements in exactly one of Aor B.(Not to be confused with delta, Δ, described below.) {1,5,6,8} ∆ {2,5,8} = {1,2,6} ∖set-theoretic complementminus;withoutset theoryA ∖ B means the set that contains all those elements of A that are not in B.[7](− can also be used for set-theoretic complement as described above.) {1,2,3,4} ∖ {3,4,5,6} = {1,2} →functionarrowfrom … toset theory, type theoryf: X → Y means the function f maps the set X into the set Y. Let f: ℤ → ℕ∪{0} be defined by f(x) := x2. ↦functionarrowmaps toset theoryf: a ↦ b means the function f maps the element a to the element b. Let f: x ↦ x+1 (the successor function). ∘function compositioncomposed withset theoryf∘g is the function, such that (f∘g)(x) = f(g(x)).[8]if f(x) := 2x, and g(x) := x + 3, then (f∘g)(x) = 2(x + 3). oHadamard productentrywise productlinear algebraFor two matrices (or vectors) of the same dimensions the Hadamard product is a matrix of the same dimensions with elements given by . This is often used in matrix based programming such as MATLABwhere the operation is done by A.*B
