List of mathematical symbols - wikipedia List of mathematical symbols This list is incomplete ; you can help by expanding it. This is a list of symbols found in all branches of mathematics to express a formula or to represent a constant. A mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept ( ultimately an arbitrary choice made as a result of the cumulative history of mathematics ), but in some situations, a different convention may be used. For example, depending on context, the triple bar `` ≡ '' may represent congruence or a definition. Further, in mathematical logic, numerical equality is sometimes represented by `` ≡ '' instead of `` = '', with the latter representing equality of well - formed formulas. In short, convention dictates the meaning. Each symbol is shown both in HTML, whose display depends on the browser 's access to an appropriate font installed on the particular device, and typeset as an image using TeX. Contents 1 Guide 2 Basic symbols 3 Symbols based on equality 4 Symbols that point left or right 5 Brackets 6 Other non-letter symbols 7 Letter - based symbols 7.1 Letter modifiers 7.2 Symbols based on Latin letters 7.3 Symbols based on Hebrew or Greek letters 8 Variations 9 See also 10 References 11 External links Guide This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. For a related list organized by mathematical topic, see List of mathematical symbols by subject. That list also includes LaTeX and HTML markup, and Unicode code points for each symbol ( note that this article does n't have the latter two, but they could certainly be added ). There is a Wikibooks guide for using maths in LaTeX, and a comprehensive LaTeX symbol list. It is also possible to check to see if a Unicode code point is available as a LaTeX command, or vice versa. Also note that where there is no LaTeX command natively available for a particular symbol ( although there may be options that require adding packages ), the symbol could be added via other options, such as setting the document up to support Unicode, and entering the character in a variety of ways ( e.g. copying and pasting, keyboard shortcuts, the \unicode ( < insertcodepoint > ) command ) as well as other options and extensive additional information. Basic symbols : Symbols widely used in mathematics, roughly through first - year calculus. More advanced meanings are included with some symbols listed here. Symbols based on equality `` = '' : Symbols derived from or similar to the equal sign, including double - headed arrows. Not surprisingly these symbols are often associated with an equivalence relation. Symbols that point left or right : Symbols, such as < and >, that appear to point to one side or another. Brackets : Symbols that are placed on either side of a variable or expression, such as x. Other non-letter symbols : Symbols that do not fall in any of the other categories. Letter - based symbols : Many mathematical symbols are based on, or closely resemble, a letter in some alphabet. This section includes such symbols, including symbols that resemble upside - down letters. Many letters have conventional meanings in various branches of mathematics and physics. These are not listed here. The See also section, below, has several lists of such usages. Letter modifiers : Symbols that can be placed on or next to any letter to modify the letter 's meaning. Symbols based on Latin letters, including those symbols that resemble or contain an X Symbols based on Hebrew or Greek letters e.g. ב, א, δ, Δ, π, Π, σ, Σ, Φ. Note : symbols resembling Λ are grouped with `` V '' under Latin letters. Variations : Usage in languages written right - to - left Basic symbols Symbol in HTML Symbol in TeX Name Explanation Examples Read as Category + + ( displaystyle + ) addition plus ; add arithmetic 4 + 6 means the sum of 4 and 6. 2 + 7 = 9 disjoint union the disjoint union of... and... set theory A + A means the disjoint union of sets A and A. A = ( 3, 4, 5, 6 ) ∧ A = ( 7, 8, 9, 10 ) ⇒ A + A = ( ( 3, 1 ), ( 4, 1 ), ( 5, 1 ), ( 6, 1 ), ( 7, 2 ), ( 8, 2 ), ( 9, 2 ), ( 10, 2 ) ) − − ( displaystyle - ) subtraction minus ; take ; subtract arithmetic 36 − 11 means the subtraction of 11 from 36. 36 − 11 = 25 negative sign negative ; minus ; the opposite of arithmetic − 3 means the additive inverse of the number 3. − ( − 5 ) = 5 set - theoretic complement minus ; without set theory A − 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 ) ± ± ( displaystyle pm ) pm plus - minus plus or minus arithmetic 6 ± 3 means both 6 + 3 and 6 − 3. The equation x = 5 ± √ 4, has two solutions, x = 7 and x = 3. plus - minus plus or minus measurement 10 ± 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. ∓ ∓ ( displaystyle mp ) mp minus - plus minus or plus arithmetic 6 ± ( 3 ∓ 5 ) means 6 + ( 3 − 5 ) and 6 − ( 3 + 5 ). cos ( x ± y ) = cos ( x ) cos ( y ) ∓ sin ( x ) sin ( y ). × ⋅ × ( displaystyle \times ) \times ⋅ ( displaystyle cdot ) cdot multiplication times ; multiplied by arithmetic 3 × 4 or 3 ⋅ 4 means the multiplication of 3 by 4. 7 ⋅ 8 = 56 dot product scalar product dot linear algebra vector algebra u ⋅ v means the dot product of vectors u and v ( 1, 2, 5 ) ⋅ ( 3, 4, − 1 ) = 6 cross product vector product cross linear algebra vector algebra u × v means the cross product of vectors u and v ( 1, 2, 5 ) × ( 3, 4, − 1 ) = i j k 5 − 1 = ( − 22, 16, − 2 ) placeholder ( silent ) functional analysis A means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. ÷ ⁄ ÷ ( displaystyle div ) div / ( displaystyle / ) division ( Obelus ) divided by ; over arithmetic 6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = 0.5 12 ⁄ 4 = 3 quotient group mod group theory G / H means the quotient of group G modulo its subgroup H. ( 0, a, 2a, b, b + a, b + 2a ) / ( 0, b ) = ( ( 0, b ), ( a, b + a ), ( 2a, b + 2a ) ) quotient set mod set theory A / ~ means the set of all ~ equivalence classes in A. If we define ~ by x ~ y ⇔ x − y ∈ Z, then R / ~ = ( x + n : n ∈ Z, x ∈ ( 0, 1 ) ). √ √ ( displaystyle surd ) surd x ( displaystyle ( sqrt ( x ) ) ) sqrt ( x ) square root ( radical symbol ) the ( principal ) square root of real numbers √ x means the nonnegative number whose square is x. √ 4 = 2 complex square root the ( complex ) square root of complex numbers If z = r exp ( iφ ) is represented in polar coordinates with − π < φ ≤ π, then √ z = √ r exp ( iφ / 2 ). √ − 1 = i ∑ ∑ ( displaystyle sum ) sum summation sum over... from... to... of calculus ∑ k = 1 n a k ( displaystyle sum _ ( k = 1 ) ^ ( n ) ( a_ ( k ) ) ) means a 1 + a 2 + ⋯ + a n ( displaystyle a_ ( 1 ) + a_ ( 2 ) + cdots + a_ ( n ) ). ∑ k = 1 4 k 2 = 1 2 + 2 2 + 3 2 + 4 2 = 1 + 4 + 9 + 16 = 30 ( displaystyle sum _ ( k = 1 ) ^ ( 4 ) ( k ^ ( 2 ) ) = 1 ^ ( 2 ) + 2 ^ ( 2 ) + 3 ^ ( 2 ) + 4 ^ ( 2 ) = 1 + 4 + 9 + 16 = 30 ) ∫ ∫ ( displaystyle int ) int indefinite integral or antiderivative indefinite integral of - OR - the antiderivative of calculus ∫ f ( x ) dx means a function whose derivative is f. ∫ x 2 d x = x 3 3 + C ( displaystyle int x ^ ( 2 ) dx = ( \frac ( x ^ ( 3 ) ) ( 3 ) ) + C ) definite integral integral from... to... of... with respect to calculus ∫ b a f ( x ) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. ∫ b a x dx = b − a / 3 line integral line / path / curve / integral of... along... calculus ∫ C f ds means the integral of f along the curve C, ∫ b a f ( r ( t ) ) r ' ( t ) dt, where r is a parametrization of C. ( If the curve is closed, the symbol ∮ may be used instead, as described below. ) ∮ ∮ ( displaystyle oint ) oint Contour integral ; closed line integral contour integral of calculus Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss 's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∮∮ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∮∮∮. The contour integral can also frequently be found with a subscript capital letter C, ∮, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss 's Law, a subscript capital S, ∮, is used to denote that the integration is over a closed surface. If C is a Jordan curve about 0, then ∮ 1 / z dz = 2πi.... ⋯ ⋮ ⋰ ⋱... ( displaystyle ldots ) ldots ⋯ ( displaystyle cdots ) cdots ⋮ ( displaystyle \vdots ) \vdots ⋱ ( displaystyle ddots ) ddots ellipsis and so forth everywhere Indicates omitted values from a pattern. 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ⋯ = 1 ∴ ∴ ( displaystyle \therefore ) \therefore therefore therefore ; so ; hence everywhere Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal. ∵ ∵ ( displaystyle \because ) \because because because ; since everywhere Sometimes used in proofs before reasoning. 11 is prime ∵ it has no positive integer factors other than itself and one. ! ! ( displaystyle ! ) factorial factorial combinatorics n ! means the product 1 × 2 ×... × n. 4 ! = 1 × 2 × 3 × 4 = 24 ( displaystyle 4 ! = 1 \times 2 \times 3 \times 4 = 24 ) logical negation not propositional logic 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. ( The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation ¬ A is preferred. ) ! ( ! A ) ⇔ A x ≠ y ⇔ ! ( x = y ) ¬  ̃ ¬ ( displaystyle \neg ) \neg ∼ ( displaystyle sim ) logical negation not propositional logic 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. ( 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 ) ⇔ A x ≠ y ⇔ ¬ ( x = y ) ∝ ∝ ( displaystyle propto ) propto proportionality is proportional to ; varies as everywhere y ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x. ∞ ∞ ( displaystyle infty ) infty infinity infinity numbers ∞ is an element of the extended number line that is greater than all real numbers ; it often occurs in limits. lim x → 0 1 x = ∞ ( displaystyle lim _ ( x \to 0 ) ( \frac ( 1 ) ( x ) ) = infty ) ∎ ▮ ‣ ◼ ( displaystyle \blacksquare ) \blacksquare ◻ ( displaystyle Box ) Box ▸ ( displaystyle \blacktriangleright ) \blacktriangleright end of proof QED ; tombstone ; Halmos finality symbol everywhere Used to mark the end of a proof. ( May also be written Q.E.D. ) Symbols based on equality Symbol in HTML Symbol in TeX Name Explanation Examples Read as Category = = ( displaystyle = ) equality is equal to ; equals everywhere x = y ( displaystyle x = y ) means x ( displaystyle x ) and y ( displaystyle y ) represent the same math object ( Both symbols have the same value ). 2 = 2 ( displaystyle 2 = 2 ) 1 + 1 = 2 ( displaystyle 1 + 1 = 2 ) 36 − 5 = 31 ( displaystyle 36 - 5 = 31 ) ≠ ≠ ( displaystyle \neq ) \ne inequality is not equal to ; does not equal everywhere x ≠ y ( displaystyle x \neq y ) means that x ( displaystyle x ) and y ( displaystyle y ) do not represent the same math object ( Both symbols do not have the same value ). ( The forms ! =, / = or < > are generally used in programming languages where ease of typing and use of ASCII text is preferred. ) 2 + 2 ≠ 5 ( displaystyle 2 + 2 \neq 5 ) 36 − 5 ≠ 30 ( displaystyle 36 - 5 \neq 30 ) ≈ ≈ ( displaystyle approx ) approx approximately equal is approximately equal to everywhere x ≈ y means x is approximately equal to y. This may also be written ≃, ≅, ~, ♎ ( Libra Symbol ), or ≒. π ≈ 3.14159 isomorphism is isomorphic to group theory G ≈ H means that group G is isomorphic ( structurally identical ) to group H. ( ≅ can also be used for isomorphic, as described below. ) Q / C ≈ V ~ ∼ ( displaystyle sim ) sim probability distribution has distribution statistics X ~ D, means the random variable X has the probability distribution D. X ~ N ( 0, 1 ), the standard normal distribution row equivalence is row equivalent to matrix theory A ~ B means that B can be generated by using a series of elementary row operations on A ( 1 2 2 4 ) ∼ ( 1 2 0 0 ) ( displaystyle ( \begin ( bmatrix ) 1&2 2&4 end ( bmatrix ) ) sim ( \begin ( bmatrix ) 1&2 0&0 end ( bmatrix ) ) ) same order of magnitude roughly similar ; poorly approximates ; is on the order of approximation theory m ~ n means the quantities m and n have the same order of magnitude, or general size. ( Note that ~ is used for an approximation that is poor, otherwise use ≈. ) 2 ~ 5 8 × 9 ~ 100 but π ≈ 10 similarity is similar to geometry △ ABC ~ △ DEF means triangle ABC is similar to ( has the same shape ) triangle DEF. asymptotically equivalent is asymptotically equivalent to asymptotic analysis f ~ g means lim n → ∞ f ( n ) g ( n ) = 1 ( displaystyle lim _ ( n \to infty ) ( \frac ( f ( n ) ) ( g ( n ) ) ) = 1 ). x ~ x + 1 equivalence relation are in the same equivalence class everywhere a ~ b means b ∈ ( a ) ( displaystyle b in ( a ) ) ( and equivalently a ∈ ( b ) ( displaystyle a in ( b ) ) ). 1 ~ 5 mod 4 = : : = ≡ : ⇔ ≜ ≝ ≐ = : ( displaystyle = : ) : = ( displaystyle : = ) ≡ ( displaystyle equiv ) equiv : ⇔ ( displaystyle : Leftrightarrow ) : Leftrightarrow ≜ ( displaystyle \triangleq ) \triangleq = d e f ( displaystyle ( overset ( \underset ( mathrm ( def ) ) ( ) ) ( = ) ) ) overset ( \underset ( mathrm ( def ) ) ( ) ) ( = ) ≐ ( displaystyle doteq ) doteq definition is defined as ; is equal by definition to everywhere x : = 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 equivalent to Q. cosh ⁡ x : = e x + e − x 2 ( displaystyle cosh x : = ( \frac ( e ^ ( x ) + e ^ ( - x ) ) ( 2 ) ) ) ( a, b ) : = a ⋅ b − b ⋅ a ( displaystyle ( a, b ) : = a cdot b-b cdot a ) ≅ ≅ ( displaystyle cong ) cong congruence is congruent to geometry △ ABC ≅ △ DEF means triangle ABC is congruent to ( has the same measurements as ) triangle DEF. isomorphic is isomorphic to abstract algebra G ≅ H means that group G is isomorphic ( structurally identical ) to group H. ( ≈ can also be used for isomorphic, as described above. ) V ≅ C × C ≡ ≡ ( displaystyle equiv ) equiv congruence relation... is congruent to... modulo... modular arithmetic a ≡ b ( mod n ) means a − b is divisible by n 5 ≡ 2 ( mod 3 ) ⇔ ↔ ⇔ ( displaystyle Leftrightarrow ) Leftrightarrow ⟺ ( displaystyle iff ) iff ↔ ( displaystyle leftrightarrow ) leftrightarrow material equivalence if and only if ; iff propositional logic 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 : = = : : = ( displaystyle : = ) = : ( displaystyle = : ) Assignment is defined to be everywhere A : = b means A is defined to have the value b. Let a : = 3, then... f ( x ) : = x + 3 Symbols that point left or right Symbol in HTML Symbol in TeX Name Explanation Examples Read as Category < > < ( displaystyle < ) > ( displaystyle > ) strict inequality is less than, is greater than order theory x < y ( displaystyle x < y ) means x is less than y. x > y ( displaystyle x > y ) means x is greater than y. 3 < 4 ( displaystyle 3 < 4 ) 5 > 4 ( displaystyle 5 > 4 ) proper subgroup is a proper subgroup of group theory H < G ( displaystyle H < G ) means H is a proper subgroup of G. 5 Z < Z ( displaystyle 5 mathrm ( Z ) < mathrm ( Z ) ) A 3 < S 3 ( displaystyle mathrm ( A ) _ ( 3 ) < mathrm ( S ) _ ( 3 ) ) ≪ ≫ ≪ ( displaystyle ll ) ≫ ( displaystyle gg ) ll gg significant ( strict ) inequality is much less than, is much greater than order theory x ≪ y means x is much less than y. x ≫ y means x is much greater than y. 0.003 ≪ 1000000 asymptotic comparison is of smaller order than, is of greater order than analytic number theory f ≪ 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 ≪ e absolute continuity is absolutely continuous with respect to measure theory μ ≪ ν ( displaystyle mu ll \nu ) means that μ ( displaystyle mu ) is absolutely continuous with respect to ν ( displaystyle \nu ), i.e., whenever ν ( A ) = 0 ( displaystyle \nu ( A ) = 0 ), we have μ ( A ) = 0 ( displaystyle mu ( A ) = 0 ). If c ( displaystyle c ) is the counting measure on ( 0, 1 ) ( displaystyle ( 0, 1 ) ) and μ ( displaystyle mu ) is the Lebesgue measure, then μ ≪ c ( displaystyle mu ll c ). ≤ ≥ ≤ ( displaystyle leq ) ≥ ( displaystyle geq ) le ge inequality is less than or equal to, is greater than or equal to order theory x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y. ( The forms < = and > = are generally used in programming languages, where ease of typing and use of ASCII text is preferred. ) ( ≦ and ≧ are also used by some writers to mean the same thing as ≤ and ≥, but this usage seems to be less common. ) 3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5 subgroup is a subgroup of group theory H ≤ G means H is a subgroup of G. Z ≤ Z A ≤ S reduction is reducible to computational complexity theory A ≤ B means the problem A can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction. If ∃ f ∈ F. ∀ x ∈ N. x ∈ A ⇔ f ( x ) ∈ B ( displaystyle exists f in F ( mbox (. ) ) \forall x in mathbb ( N ) ( mbox (. ) ) x in A Leftrightarrow f ( x ) in B ) then A ≤ F B ( displaystyle A leq _ ( F ) B ) ≦ ≧ ≦ ( displaystyle leqq ) ≧ ( displaystyle geqq ) leqq geqq congruence relation... is less than... is greater than... modular arithmetic 10a ≡ 5 ( mod 5 ) for 1 ≦ a ≦ 10 vector inequality... is less than or equal... is greater than or equal... order theory x ≦ y means that each component of vector x is less than or equal to each corresponding component of vector y. x ≧ y means that each component of vector x is greater than or equal to each corresponding component of vector y. It is important to note that x ≦ y remains true if every element is equal. However, if the operator is changed, x ≤ y is true if and only if x ≠ y is also true. ≺ ≻ ≺ ( displaystyle prec ) ≻ ( displaystyle succ ) prec succ Karp reduction is Karp reducible to ; is polynomial - time many - one reducible to computational complexity theory L ≺ L means that the problem L is Karp reducible to L. If L ≺ L and L ∈ P, then L ∈ P. Nondominated order is nondominated by Multi-objective optimization P ≺ Q means that the element P is nondominated by element Q. If P ≺ Q then ∀ i P i ≤ Q i ∧ ∃ P i < Q i ( displaystyle \forall _ ( i ) P_ ( i ) leq Q_ ( i ) land exists P_ ( i ) < Q_ ( i ) ) ◅ ▻ & # x25C5 ; & # x25BB ; ◃ ( displaystyle \triangleleft ) ▹ ( displaystyle \triangleright ) \triangleleft \triangleright normal subgroup is a normal subgroup of group theory N ◅ G means that N is a normal subgroup of group G. Z ( G ) ◅ G ideal is an ideal of ring theory I ◅ R means that I is an ideal of ring R. ( 2 ) ◅ Z antijoin the antijoin of relational algebra R ▻ S means the antijoin of the relations R and S, the tuples in R for which there is not a tuple in S that is equal on their common attribute names. R ▹ S = R − R ⋉ S ( displaystyle R \triangleright S = R-R ltimes S ) ⇒ → ⊃ ⇒ ( displaystyle Rightarrow ) → ( displaystyle \rightarrow ) ⊃ ( displaystyle supset ) Rightarrow \rightarrow supset material implication implies ; if... then propositional logic, Heyting algebra 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 = 6 ⇒ x − 5 = 36 − 5 = 31 is true, but x − 5 = 36 − 5 = 31 ⇒ x = 6 is in general false ( since x could be − 6 ). ⊆ ⊂ ⊆ ( displaystyle subseteq ) ⊂ ( displaystyle subset ) subseteq subset subset is a subset of set theory ( subset ) A ⊆ B means every element of A is also an element of B. ( proper subset ) A ⊂ B means A ⊆ B but A ≠ B. ( Some writers use the symbol ⊂ as if it were the same as ⊆. ) ( A ∩ B) ⊆ A N ⊂ Q Q ⊂ R ⊇ ⊃ ⊇ ( displaystyle supseteq ) ⊃ ( displaystyle supset ) supseteq supset superset is a superset of set theory A ⊇ 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) ⊇ B R ⊃ Q → → ( displaystyle \to ) \to function arrow from... to set theory, type theory f : X → Y means the function f maps the set X into the set Y. Let f : Z → N ∪ ( 0 ) be defined by f ( x ) : = x. ↦ ↦ ( displaystyle mapsto ) mapsto function arrow maps to set theory f : a ↦ b means the function f maps the element a to the element b. Let f : x ↦ x + 1 ( the successor function ). ← ← ( displaystyle leftarrow ) leftarrow Converse implication... if... logic a ← b means that for the propositions a and b, if b implies a, then a is the converse implication of b. a to the element b. This reads as `` a if b '', or `` not b without a ''. It is not to be confused with the assignment operator in computer science. < : < < : ( displaystyle < : ) < ⋅ ( displaystyle ( < ) ( cdot ) ) subtype is a subtype of type theory T < : T means that T is a subtype of T. If S < : T and T < : U then S < : U ( transitivity ). cover is covered by order theory x < y means that x is covered by y. ( 1, 8 ) < ( 1, 3, 8 ) among the subsets of ( 1, 2,..., 10 ) ordered by containment. ⊧ ⊨ ( displaystyle \vDash ) \vDash entailment entails model theory A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. A ⊧ A ∨ ¬ A ⊢ ⊢ ( displaystyle \vdash ) \vdash inference infers ; is derived from propositional logic, predicate logic x ⊢ y means y is derivable from x. A → B ⊢ ¬ B → ¬ A partition is a partition of number theory p ⊢ n means that p is a partition of n. ( 4, 3, 1, 1 ) ⊢ 9, ∑ λ ⊢ n ( f λ ) 2 = n ! ( displaystyle sum _ ( lambda \vdash n ) ( f_ ( lambda ) ) ^ ( 2 ) = n ! ) ⟨ ⟨ ( displaystyle langle ) langle bra vector the bra... ; the dual of... Dirac notation ⟨ φ means the dual of the vector φ ⟩, a linear functional which maps a ket ψ ⟩ onto the inner product ⟨ φ ψ ⟩. ⟩ ⟩ ( displaystyle \rangle ) \rangle ket vector the ket... ; the vector... Dirac notation φ ⟩ means the vector with label φ, which is in a Hilbert space. A qubit 's state can be represented as α 0 ⟩ + β 1 ⟩, where α and β are complex numbers s.t. α + β = 1. Brackets Symbol in HTML Symbol in TeX Name Explanation Examples Read as Category ( ) ( displaystyle ( choose ) ) ( choose ) combination ; binomial coefficient n choose k combinatorics ( n k ) = n ! / ( n − k ) ! k ! = ( n − k + 1 ) ⋯ ( n − 2 ) ⋅ ( n − 1 ) ⋅ n k ! ( displaystyle ( \begin ( pmatrix ) n k end ( pmatrix ) ) = ( \frac ( n ! / ( n-k ) ! ) ( k ! ) ) = ( \frac ( ( n - k + 1 ) cdots ( n - 2 ) cdot ( n - 1 ) cdot n ) ( k ! ) ) ) means ( in the case of n = positive integer ) the number of combinations of k elements drawn from a set of n elements. ( This may also be written as C ( n, k ), C ( n ; k ), C, C, or ⟨ n k ⟩ ( displaystyle left langle ( \begin ( matrix ) n k end ( matrix ) ) \right \rangle ). ) ( 36 5 ) = 36 ! / ( 36 − 5 ) ! 5 ! = 32 ⋅ 33 ⋅ 34 ⋅ 35 ⋅ 36 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 = 376992 ( displaystyle ( \begin ( pmatrix ) 36 5 end ( pmatrix ) ) = ( \frac ( 36 ! / ( 36 - 5 ) ! ) ( 5 ! ) ) = ( \frac ( 32 cdot 33 cdot 34 cdot 35 cdot 36 ) ( 1 cdot 2 cdot 3 cdot 4 cdot 5 ) ) = 376992 ) (. 5 7 ) = − 5.5 ⋅ − 4.5 ⋅ − 3.5 ⋅ − 2.5 ⋅ − 1.5 ⋅ −. 5 ⋅. 5 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 = 33 2048 ( displaystyle ( \begin ( pmatrix ). 5 7 end ( pmatrix ) ) = ( \frac ( - 5.5 cdot - 4.5 cdot - 3.5 cdot - 2.5 cdot - 1.5 cdot -. 5 cdot. 5 ) ( 1 cdot 2 cdot 3 cdot 4 cdot 5 cdot 6 cdot 7 ) ) = ( \frac ( 33 ) ( 2048 ) ) , ! ) ( ( ) ) ( displaystyle left ( ! ! ( choose ) ! ! \right ) ) left ( ! ! ( choose ) ! ! \right ) multiset coefficient u multichoose k combinatorics ( ( u k ) ) = ( u + k − 1 k ) = ( u + k − 1 ) ! / ( u − 1 ) ! k ! ( displaystyle left ( ! ! ( u choose k ) ! ! \right ) = ( u + k - 1 choose k ) = ( \frac ( ( u + k - 1 ) ! / ( u-1 ) ! ) ( k ! ) ) ) ( when u is positive integer ) means reverse or rising binomial coefficient. ( ( − 5.5 7 ) ) = − 5.5 ⋅ − 4.5 ⋅ − 3.5 ⋅ − 2.5 ⋅ − 1.5 ⋅ −. 5 ⋅. 5 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 = (. 5 7 ) = 33 2048 ( displaystyle left ( ! ! ( - 5.5 choose 7 ) ! ! \right ) = ( \frac ( - 5.5 cdot - 4.5 cdot - 3.5 cdot - 2.5 cdot - 1.5 cdot -. 5 cdot. 5 ) ( 1 cdot 2 cdot 3 cdot 4 cdot 5 cdot 6 cdot 7 ) ) = (. 5 choose 7 ) = ( \frac ( 33 ) ( 2048 ) ) , ! )...... ( displaystyle ldots ! , ) ldots ! , absolute value ; modulus absolute value of ; modulus of numbers 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 magnitude Euclidean norm of geometry x means the ( Euclidean ) length of vector x. For x = ( 3, − 4 ) x = 3 2 + ( − 4 ) 2 = 5 ( displaystyle ( \textbf ( x ) ) = ( sqrt ( 3 ^ ( 2 ) + ( - 4 ) ^ ( 2 ) ) ) = 5 ) determinant determinant of matrix theory A means the determinant of the matrix A 1 2 2 9 = 5 ( displaystyle ( \begin ( vmatrix ) 1&2 2&9 end ( vmatrix ) ) = 5 ) cardinality cardinality of ; size of ; order of set theory X means the cardinality of the set X. ( # may be used instead as described below. ) ( 3, 5, 7, 9 ) = 4. ‖... ‖ ∥... ∥ ( displaystyle ldots ! , ) ldots ! , norm norm of ; length of linear algebra ‖ x ‖ means the norm of the element x of a normed vector space. ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ nearest integer function nearest integer to numbers ‖ x ‖ means the nearest integer to x. ( This may also be written ( x ), ⌊ x ⌉, nint ( x ) or Round ( x ). ) ‖ 1 ‖ = 1, ‖ 1.6 ‖ = 2, ‖ − 2.4 ‖ = − 2, ‖ 3.49 ‖ = 3 (, ) (, ) ( displaystyle ( ( , ! ) ) ! , ) ( ( , ! ) ) ! , set brackets the set of... set theory ( a, b, c ) means the set consisting of a, b, and c. N = ( 1, 2, 3,... ) ( : ) ( ) ( ; ) ( : ) ( displaystyle ( :) ! , ) ( :) ! , ( ) ( displaystyle ( ) ! , ) ( ) ! , ( ; ) ( displaystyle ( ; ) ! , ) ( ; ) ! , set builder notation the set of... such that 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 : n < 20 ) = ( 1, 2, 3, 4 ) ⌊... ⌋ ⌊... ⌋ ( displaystyle lfloor ldots \rfloor ! , ) lfloor ldots \rfloor ! , floor floor ; greatest integer ; entier numbers ⌊ x ⌋ means the floor of x, i.e. the largest integer less than or equal to x. ( This may also be written ( x ), floor ( x ) or int ( x ). ) ⌊ 4 ⌋ = 4, ⌊ 2.1 ⌋ = 2, ⌊ 2.9 ⌋ = 2, ⌊ − 2.6 ⌋ = − 3 ⌈... ⌉ ⌈... ⌉ ( displaystyle lceil ldots \rceil ! , ) lceil ldots \rceil ! , ceiling ceiling numbers ⌈ x ⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x. ( This may also be written ceil ( x ) or ceiling ( x ). ) ⌈ 4 ⌉ = 4, ⌈ 2.1 ⌉ = 3, ⌈ 2.9 ⌉ = 3, ⌈ − 2.6 ⌉ = − 2 ⌊... ⌉ ⌊... ⌉ ( displaystyle lfloor ldots \rceil ! , ) lfloor ldots \rceil ! , nearest integer function nearest integer to numbers ⌊ x ⌉ means the nearest integer to x. ( This may also be written ( x ), x, nint ( x ) or Round ( x ). ) ⌊ 2 ⌉ = 2, ⌊ 2.6 ⌉ = 3, ⌊ − 3.4 ⌉ = − 3, ⌊ 4.49 ⌉ = 4 ( : ) ( : ) ( displaystyle ( :) ! , ) ( :) ! , degree of a field extension the degree of field theory ( K : F ) means the degree of the extension K : F. ( Q ( √ 2 ) : Q ) = 2 ( C : R ) = 2 ( R : Q ) = ∞ ( ) (, ) (,, ) ( ) ( displaystyle ( ) ! , ) ( ) ! , (, ) ( displaystyle ( , ) ! , ) ( , ) ! , (,, ) ( displaystyle ( , , ) ! , ) equivalence class the equivalence class of abstract algebra ( a ) means the equivalence class of a, i.e. ( x : x ~ a ), where ~ is an equivalence relation. ( a ) means the same, but with R as the equivalence relation. Let a ~ b be true iff a ≡ b ( mod 5 ). Then ( 2 ) = (..., − 8, − 3, 2, 7,... ). floor floor ; greatest integer ; entier numbers ( x ) means the floor of x, i.e. the largest integer less than or equal to x. ( This may also be written ⌊ x ⌋, floor ( x ) or int ( x ). Not to be confused with the nearest integer function, as described below. ) ( 3 ) = 3, ( 3.5 ) = 3, ( 3.99 ) = 3, ( − 3.7 ) = − 4 nearest integer function nearest integer to numbers ( x ) means the nearest integer to x. ( This may also be written ⌊ x ⌉, x, nint ( x ) or Round ( x ). Not to be confused with the floor function, as described above. ) ( 2 ) = 2, ( 2.6 ) = 3, ( − 3.4 ) = − 3, ( 4.49 ) = 4 Iverson bracket 1 if true, 0 otherwise propositional logic ( S ) maps a true statement S to 1 and a false statement S to 0. ( 0 = 5 ) = 0, ( 7 > 0 ) = 1, ( 2 ∈ ( 2, 3, 4 ) ) = 1, ( 5 ∈ ( 2, 3, 4 ) ) = 0 image image of... under... everywhere f ( X ) means ( f ( x ) : x ∈ X ), the image of the function f under the set X ⊆ dom ( f ). ( This may also be written as f ( X ) if there is no risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain. ) sin ⁡ ( R ) = ( − 1, 1 ) ( displaystyle sin ( mathbb ( R ) ) = ( - 1, 1 ) ) closed interval closed interval order theory ( a, b ) = ( x ∈ R : a ≤ x ≤ b ) ( displaystyle ( a, b ) = ( x in mathbb ( R ) : a leq x leq b ) ). 0 and 1 / 2 are in the interval ( 0, 1 ). commutator the commutator of group theory, ring theory ( g, h ) = g h gh ( or ghg h ), if g, h ∈ G ( a group ). ( a, b ) = ab − ba, if a, b ∈ R ( a ring or commutative algebra ). x = x ( x, y ) ( group theory ). ( AB, C ) = A ( B, C ) + ( A, C ) B ( ring theory ). triple scalar product the triple scalar product of vector calculus ( a, b, c ) = a × b c, the scalar product of a × b with c. ( a, b, c ) = ( b, c, a ) = ( c, a, b ). ( ) (, ) ( ) ( displaystyle ( ) ! , ) ( ) ! , (, ) ( displaystyle ( , ) ! , ) ( , ) ! , function application of set theory f ( x ) means the value of the function f at the element x. If f ( x ) : = x − 5, then f ( 6 ) = 6 − 5 = 36 − 5 = 31. image image of... under... everywhere f ( X ) means ( f ( x ) : x ∈ X ), the image of the function f under the set X ⊆ dom ( f ). ( This may also be written as f ( X ) if there is a risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain. ) sin ⁡ ( R ) = ( − 1, 1 ) ( displaystyle sin ( mathbb ( R ) ) = ( - 1, 1 ) ) precedence grouping parentheses everywhere Perform the operations inside the parentheses first. ( 8 / 4 ) / 2 = 2 / 2 = 1, but 8 / ( 4 / 2 ) = 8 / 2 = 4. tuple tuple ; n - tuple ; ordered pair / triple / etc ; row vector ; sequence everywhere An ordered list ( or sequence, or horizontal vector, or row vector ) of values. ( Note that the notation ( a, b ) is ambiguous : it could be an ordered pair or an open interval. Set theorists and computer scientists often use angle brackets ⟨ ⟩ instead of parentheses. ) ( a, b ) is an ordered pair ( or 2 - tuple ). ( a, b, c ) is an ordered triple ( or 3 - tuple ). ( ) is the empty tuple ( or 0 - tuple ). highest common factor highest common factor ; greatest common divisor ; hcf ; gcd number theory ( a, b ) means the highest common factor of a and b. ( This may also be written hcf ( a, b ) or gcd ( a, b ). ) ( 3, 7 ) = 1 ( they are coprime ) ; ( 15, 25 ) = 5. (, ) ), ( (, ) ( displaystyle ( , ) ! , ) ( , ) ! , ( , ) ! , ), ( ( displaystyle ) , ( ! , ) ) , ( ! , ) open interval open interval order theory ( a, b ) = ( x ∈ R : a < x < b ) ( displaystyle ( a, b ) = ( x in mathbb ( R ) : a < x < b ) ). ( Note that the notation ( a, b ) is ambiguous : it could be an ordered pair or an open interval. The notation ) a, b ( can be used instead. ) 4 is not in the interval ( 4, 18 ). ( 0, + ∞ ) equals the set of positive real numbers. (, ) ), ) (, ) ( displaystyle ( , ) ! , ) ( , ) ! , ), ) ( displaystyle ) , ) ! , ) , ) ! , ) left - open interval half - open interval ; left - open interval order theory ( a, b ) = ( x ∈ R : a < x ≤ b ) ( displaystyle ( a, b ) = ( x in mathbb ( R ) : a < x leq b ) ). ( − 1, 7 ) and ( − ∞, − 1 ) (, ) (, ( (, ) ( displaystyle ( , ) ! , ) ( , ) ! , (, ( ( displaystyle ( , ( ! , ) ( , ( ! , right - open interval half - open interval ; right - open interval order theory ( a, b ) = ( x ∈ R : a ≤ x < b ) ( displaystyle ( a, b ) = ( x in mathbb ( R ) : a leq x < b ) ). ( 4, 18 ) and ( 1, + ∞ ) ⟨ ⟩ ⟨, ⟩ ⟨ ⟩ ( displaystyle langle \rangle ! , ) langle \rangle ! , ⟨, ⟩ ( displaystyle langle , \rangle ! , ) langle , \rangle ! , inner product inner product of linear algebra ⟨ u, v ⟩ means the inner product of u and v, where u and v are members of an inner product space. Note that the notation ⟨ u, v ⟩ may be ambiguous : it could mean the inner product or the linear span. There are many variants of the notation, such as ⟨ u v ⟩ and ( u v ), which are described below. For spatial vectors, the dot product notation, x y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more `` keyboard friendly '' forms < and > are sometimes seen. These are avoided in mathematical texts. The standard inner product between two vectors x = ( 2, 3 ) and y = ( − 1, 5 ) is : ⟨ x, y ⟩ = 2 × − 1 + 3 × 5 = 13 average average of statistics let S be a subset of N for example, ⟨ S ⟩ ( displaystyle langle S \rangle ) represents the average of all the elements in S. for a time series : g ( t ) ( t = 1, 2,... ) we can define the structure functions S ( τ ( displaystyle \tau ) ) : S q = ⟨ g ( t + τ ) − g ( t ) q ⟩ t ( displaystyle S_ ( q ) = langle g ( t+ \tau ) - g ( t ) ^ ( q ) \rangle _ ( t ) ) expectation value the expectation value of probability theory For a single discrete variable x ( displaystyle x ) of a function f ( x ) ( displaystyle f ( x ) ), the expectation value of f ( x ) ( displaystyle f ( x ) ) is defined as ⟨ f ( x ) ⟩ = ∑ x f ( x ) P ( x ) ( displaystyle langle f ( x ) \rangle = sum _ ( x ) f ( x ) P ( x ) ), and for a single continuous variable the expectation value of f ( x ) ( displaystyle f ( x ) ) is defined as ⟨ f ( x ) ⟩ = ∫ x f ( x ) P ( x ) ( displaystyle langle f ( x ) \rangle = int _ ( x ) f ( x ) P ( x ) ) ; where P ( x ) ( displaystyle P ( x ) ) is the PDF of the variable x ( displaystyle x ). linear span ( linear ) span of ; linear hull of linear algebra ⟨ S ⟩ means the span of S ⊆ V. That is, it is the intersection of all subspaces of V which contain S. ⟨ u, u,... ⟩ is shorthand for ⟨ ( u, u,... ) ⟩. Note that the notation ⟨ u, v ⟩ may be ambiguous : it could mean the inner product or the linear span. The span of S may also be written as Sp ( S ). ⟨ ( 1 0 0 ), ( 0 1 0 ), ( 0 0 1 ) ⟩ = R 3 ( displaystyle left langle left ( ( \begin ( smallmatrix ) 1 0 0 end ( smallmatrix ) ) \right ), left ( ( \begin ( smallmatrix ) 0 1 0 end ( smallmatrix ) ) \right ), left ( ( \begin ( smallmatrix ) 0 0 1 end ( smallmatrix ) ) \right ) \right \rangle = mathbb ( R ) ^ ( 3 ) ). subgroup generated by a set the subgroup generated by group theory ⟨ S ⟩ ( displaystyle langle S \rangle ) means the smallest subgroup of G ( where S ⊆ G, a group ) containing every element of S. ⟨ g 1, g 2,... ⟩ ( displaystyle left langle g_ ( 1 ), g_ ( 2 ), dots \right \rangle ) is shorthand for ⟨ ( g 1, g 2,... ) ⟩ ( displaystyle left langle left ( g_ ( 1 ), g_ ( 2 ), dots \right ) \right \rangle ). In S, ⟨ ( 1 2 ) ⟩ = ( i d, ( 1 2 ) ) ( displaystyle langle ( 1 ; 2 ) \rangle = ( id, ; ( 1 ; 2 ) ) ) and ⟨ ( 1 2 3 ) ⟩ = ( i d, ( 1 2 3 ), ( 1 2 3 ) ) ) ( displaystyle langle ( 1 ; 2 ; 3 ) \rangle = ( id, ; ( 1 ; 2 ; 3 ), ( 1 ; 2 ; 3 ) ) ) ). tuple tuple ; n - tuple ; ordered pair / triple / etc ; row vector ; sequence everywhere An ordered list ( or sequence, or horizontal vector, or row vector ) of values. ( The notation ( a, b ) is often used as well. ) ⟨ a, b ⟩ ( displaystyle langle a, b \rangle ) is an ordered pair ( or 2 - tuple ). ⟨ a, b, c ⟩ ( displaystyle langle a, b, c \rangle ) is an ordered triple ( or 3 - tuple ). ⟨ ⟩ ( displaystyle langle \rangle ) is the empty tuple ( or 0 - tuple ). ⟨ ⟩ ( ) ⟨ ⟩ ( displaystyle langle \rangle ! , ) langle \rangle ! , ( ) ( displaystyle ( ) ! , ) ( ) ! , inner product inner product of linear algebra ⟨ u v ⟩ means the inner product of u and v, where u and v are members of an inner product space. ( u v ) means the same. Another variant of the notation is ⟨ u, v ⟩ which is described above. For spatial vectors, the dot product notation, x y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more `` keyboard friendly '' forms < and > are sometimes seen. These are avoided in mathematical texts. Other non-letter symbols Symbol in HTML Symbol in TeX Name Explanation Examples Read as Category * ∗ ( displaystyle * ! , ) convolution convolution ; convolved with functional analysis f ∗ g means the convolution of f and g. ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ ( displaystyle ( f * g ) ( t ) = int _ ( 0 ) ^ ( t ) f ( \tau ) g ( t - \tau ) , d \tau ). complex conjugate conjugate complex numbers z means the complex conjugate of z. ( z  ̄ ( displaystyle ( \bar ( z ) ) ) can also be used for the conjugate of z, as described below. ) ( 3 + 4 i ) ∗ = 3 − 4 i ( displaystyle ( 3 + 4i ) ^ ( ast ) = 3 - 4i ). group of units the group of units of ring theory R 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, or U ( R ). ( Z / 5 Z ) ∗ = ( ( 1 ), ( 2 ), ( 3 ), ( 4 ) ) ≅ C 4 ( displaystyle ( \begin ( aligned ) ( mathbb ( Z ) / 5 mathbb ( Z ) ) ^ ( ast ) & = ( ( 1 ), ( 2 ), ( 3 ), ( 4 ) ) & cong mathrm ( C ) _ ( 4 ) end ( aligned ) ) ) hyperreal numbers the ( set of ) hyperreals non-standard analysis R means the set of hyperreal numbers. Other sets can be used in place of R. N is the hypernatural numbers. Hodge dual Hodge dual ; Hodge star linear algebra ∗ v means the Hodge dual of a vector v. If v is a k - vector within an n - dimensional oriented inner product space, then ∗ v is an ( n − k ) - vector. If ( e i ) ( displaystyle ( e_ ( i ) ) ) are the standard basis vectors of R 5 ( displaystyle mathbb ( R ) ^ ( 5 ) ), ∗ ( e 1 ∧ e 2 ∧ e 3 ) = e 4 ∧ e 5 ( displaystyle * ( e_ ( 1 ) wedge e_ ( 2 ) wedge e_ ( 3 ) ) = e_ ( 4 ) wedge e_ ( 5 ) ) Kleene star Kleene star computer science, mathematical logic Corresponds to the usage of * in regular expressions. If ∑ is a set of strings, then ∑ * is the set of all strings that can be created by concatenating members of ∑. The same string can be used multiple times, and the empty string is also a member of ∑ *. If ∑ = ( ' a ', ' b ', ' c ' ) then ∑ * includes ' ', ' a ', ' ab ', ' aba ', ' abac ', etc. The full set can not be enumerated here since it is countably infinite, but each individual string must have finite length. ∝ ∝ ( displaystyle propto ! , ) propto ! , proportionality is proportional to ; varies as everywhere y ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x. Karp reduction is Karp reducible to ; is polynomial - time many - one reducible to computational complexity theory A ∝ B means the problem A can be polynomially reduced to the problem B. If L ∝ L and L ∈ P, then L ∈ P. ∖ ∖ ( displaystyle setminus ! , ) setminus set - theoretic complement minus ; without ; throw out ; not set theory A ∖ B means the set that contains all those elements of A that are not in B. ( − can also be used for set - theoretic complement as described above. ) ( 1, 2, 3, 4 ) ∖ ( 3, 4, 5, 6 ) = ( 1, 2 ) ( displaystyle ! , ) conditional event given probability P ( 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 restriction restriction of... to... ; restricted to set theory f means the function f is restricted to the set A, that is, it is the function with domain A ∩ dom ( f ) that agrees with f. The function f : R → R defined by f ( x ) = x is not injective, but f is injective. such that such that ; so that everywhere 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 ). ∣ ∤ ∣ ( displaystyle mid ! , ) mid ∤ ( displaystyle \nmid ! , ) \nmid divisor, divides divides number theory a ∣ b means a divides b. a ∤ b means a does not divide b. ( The symbol ∣ can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar character is often used instead. ) Since 15 = 3 × 5, it is true that 3 ∣ 15 and 5 ∣ 15. ∣ ∣ ∣ ∣ ( displaystyle mid mid ! , ) mid mid exact divisibility exactly divides number theory p ∣ ∣ n means p exactly divides n ( i.e. p divides n but p does not ). 2 ∣ ∣ 360. ∥ ∦ ⋕ ∥ ( displaystyle ! , ) Requires the viewer to support Unicode : \unicode ( x2225 ), \unicode ( x2226 ), and \unicode ( x22D5 ). mathrel ( \rlap ( , parallel ) ) requires setmathfont ( MathJax ). parallel is parallel to geometry x ∥ y means x is parallel to y. x ∦ y means x is not parallel to y. x ⋕ y means x is equal and parallel to y. ( The symbol ∥ can be difficult to type, and its negation is rare, so two regular but slightly longer vertical bar characters are often used instead. ) If l ∥ m and m ⊥ n then l ⊥ n. incomparability is incomparable to order theory x ∥ y means x is incomparable to y. ( 1, 2 ) ∥ ( 2, 3 ) under set containment. # # ( displaystyle # ! , ) sharp cardinality cardinality of ; size of ; order of set theory # X means the cardinality of the set X. (... may be used instead as described above. ) # ( 4, 6, 8 ) = 3 connected sum connected sum of ; knot sum of ; knot composition of topology, knot theory A#B is the connected sum of the manifolds A and B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition. A#S is homeomorphic to A, for any manifold A, and the sphere S. primorial primorial number theory n # is product of all prime numbers less than or equal to n. 12 # = 2 × 3 × 5 × 7 × 11 = 2310 : : ( displaystyle :! , ) such that such that ; so that everywhere : means `` such that '', and is used in proofs and the set - builder notation ( described below ). ∃ n ∈ N : n is even. field extension extends ; over field theory K : F means the field K extends the field F. This may also be written as K ≥ F. R : Q inner product of matrices inner product of linear algebra A : 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. A : B = ∑ i, j A i j B i j ( displaystyle A : B = sum _ ( i, j ) A_ ( ij ) B_ ( ij ) ) index of a subgroup index of subgroup group theory The 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 G : H = G H ( displaystyle G : H = ( \frac ( G ) ( H ) ) ) division divided by over everywhere A : B means the division of A with B ( dividing A by B) 10 : 2 = 5 ⋮ ⋮ ( displaystyle \vdots ! , ) \vdots ! , vertical ellipsis vertical ellipsis everywhere Denotes that certain constants and terms are missing out ( e.g. for clarity ) and that only the important terms are being listed. P ( r, t ) = χ ⋮ E ( r, t 1 ) E ( r, t 2 ) E ( r, t 3 ) ( displaystyle P ( r, t ) = chi \vdots E ( r, t_ ( 1 ) ) E ( r, t_ ( 2 ) ) E ( r, t_ ( 3 ) ) ) ≀ ≀ ( displaystyle wr ! , ) wr ! , wreath product wreath product of... by... group theory A ≀ H means the wreath product of the group A by the group H. This may also be written A H. S n ≀ Z 2 ( displaystyle mathrm ( S ) _ ( n ) wr mathrm ( Z ) _ ( 2 ) ) is isomorphic to the automorphism group of the complete bipartite graph on ( n, n ) vertices. ↯ ⨳ ⇒ ⇐ \blitza lighting : requires \usepackage ( stmaryd ). smashtimes requires \usepackage ( unicode - math ) and setmathfont ( XITS Math ) or another Open Type Math Font. ⇒ ⇐ ( displaystyle Rightarrow Leftarrow ) Rightarrow Leftarrow ⊥ ( displaystyle \bot ) \bot ↮ ( displaystyle \nleftrightarrow ) \nleftrightarrow \textreferencemark Contradiction ! downwards zigzag arrow contradiction ; this contradicts that everywhere Denotes that contradictory statements have been inferred. For clarity, the exact point of contradiction can be appended. x + 4 = x − 3 ※ Statement : Every finite, non-empty, ordered set has a largest element. Otherwise, let 's assume that X ( displaystyle X ) is a finite, non-empty, ordered set with no largest element. Then, for some x 1 ∈ X ( displaystyle x_ ( 1 ) in X ), there exists an x 2 ∈ X ( displaystyle x_ ( 2 ) in X ) with x 1 < x 2 ( displaystyle x_ ( 1 ) < x_ ( 2 ) ), but then there 's also an x 3 ∈ X ( displaystyle x_ ( 3 ) in X ) with x 2 < x 3 ( displaystyle x_ ( 2 ) < x_ ( 3 ) ), and so on. Thus, x 1, x 2, x 3,... ( displaystyle x_ ( 1 ), x_ ( 2 ), x_ ( 3 ),... ) are distinct elements in X ( displaystyle X ). ↯ X ( displaystyle X ) is finite. ⊕ ⊻ ⊕ ( displaystyle oplus ! , ) oplus ! , ⊻ ( displaystyle \veebar ! , ) \veebar ! , exclusive or xor propositional logic, Boolean algebra The 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 sum direct sum of abstract algebra The direct sum is a special way of combining several objects into one general object. ( The bun symbol ⊕, or the coproduct symbol ∐, 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 ) ) ∧ ◯ ( displaystyle ( ~ wedge ! ! ! ! ! ! \bigcirc ~ ) ) ( ~ wedge ! ! ! ! ! ! \bigcirc ~ ) Kulkarni -- Nomizu product Kulkarni -- Nomizu product tensor algebra Derived from the tensor product of two symmetric type ( 0, 2 ) tensors ; it has the algebraic symmetries of the Riemann tensor. f = g ∧ ◯ h ( displaystyle f = g ( , wedge ! ! ! ! ! ! \bigcirc , ) h ) has components f α β γ δ = g α γ h β δ + g β δ h α γ − g α δ h β γ − g β γ h α δ ( displaystyle f_ ( alpha \beta gamma delta ) = g_ ( alpha gamma ) h_ ( \beta delta ) + g_ ( \beta delta ) h_ ( alpha gamma ) - g_ ( alpha delta ) h_ ( \beta gamma ) - g_ ( \beta gamma ) h_ ( alpha delta ) ). ◻ ( displaystyle Box ! , ) Box ! D'Alembertian ; wave operator non-Euclidean Laplacian vector calculus It 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. ◻ = 1 c 2 ∂ 2 ∂ t 2 − ∂ 2 ∂ x 2 − ∂ 2 ∂ y 2 − ∂ 2 ∂ z 2 ( displaystyle square = ( \frac ( 1 ) ( c ^ ( 2 ) ) ) ( partial ^ ( 2 ) over partial t ^ ( 2 ) ) - ( partial ^ ( 2 ) over partial x ^ ( 2 ) ) - ( partial ^ ( 2 ) over partial y ^ ( 2 ) ) - ( partial ^ ( 2 ) over partial z ^ ( 2 ) ) ) Letter - based symbols Includes upside - down letters. Letter modifiers Also called diacritics. Symbol in HTML Symbol in TeX Name Explanation Examples Read as Category a  ̄ ( displaystyle ( \bar ( a ) ) ) \bar ( a ) mean overbar ;... bar statistics x  ̄ ( displaystyle ( \bar ( x ) ) ) ( often read as `` x bar '' ) is the mean ( average value of x i ( displaystyle x_ ( i ) ) ). x = ( 1, 2, 3, 4, 5 ) ; x  ̄ = 3 ( displaystyle x = ( 1, 2, 3, 4, 5 ) ; ( \bar ( x ) ) = 3 ). finite sequence, tuple finite sequence, tuple model theory a  ̄ ( displaystyle ( overline ( a ) ) ) means the finite sequence / tuple ( a 1, a 2,..., a n ). ( displaystyle ( a_ ( 1 ), a_ ( 2 ),..., a_ ( n ) ). ). a  ̄ : = ( a 1, a 2,..., a n ) ( displaystyle ( overline ( a ) ) : = ( a_ ( 1 ), a_ ( 2 ),..., a_ ( n ) ) ). algebraic closure algebraic closure of field theory F  ̄ ( displaystyle ( overline ( F ) ) ) is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as Q  ̄ ( displaystyle ( overline ( mathbb ( Q ) ) ) ) because it is the algebraic closure of the rational numbers Q ( displaystyle ( mathbb ( Q ) ) ). complex conjugate conjugate complex numbers z  ̄ ( displaystyle ( overline ( z ) ) ) means the complex conjugate of z. ( z can also be used for the conjugate of z, as described above. ) 3 + 4 i  ̄ = 3 − 4 i ( displaystyle ( overline ( 3 + 4i ) ) = 3 - 4i ). topological closure ( topological ) closure of topology S  ̄ ( displaystyle ( overline ( S ) ) ) is the topological closure of the set S. This may also be denoted as cl ( S ) or Cl ( S ). In the space of the real numbers, Q  ̄ = R ( displaystyle ( overline ( mathbb ( Q ) ) ) = mathbb ( R ) ) ( the rational numbers are dense in the real numbers ). a ⇀ ( displaystyle ( overset ( \rightharpoonup ) ( a ) ) ) a ⇀ ( displaystyle ( overset ( \rightharpoonup ) ( a ) ) ) overset ( \rightharpoonup ) ( a ) vector harpoon linear algebra â a ^ ( displaystyle ( hat ( a ) ) ) hat a unit vector hat geometry a ^ ( displaystyle mathbf ( hat ( a ) ) ) ( pronounced `` a hat '' ) is the normalized version of vector a ( displaystyle mathbf ( a ) ), having length 1. estimator estimator for statistics θ ^ ( displaystyle ( hat ( \theta ) ) ) is the estimator or the estimate for the parameter θ ( displaystyle \theta ). The estimator μ ^ = ∑ i x i n ( displaystyle mathbf ( hat ( mu ) ) = ( \frac ( sum _ ( i ) x_ ( i ) ) ( n ) ) ) produces a sample estimate μ ^ ( x ) ( displaystyle mathbf ( hat ( mu ) ) ( mathbf ( x ) ) ) for the mean μ ( displaystyle mu ). ′ ′ ( displaystyle ' ) ' derivative... prime ; derivative of calculus f ′ ( x ) means the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. ( The single - quote character ' is sometimes used instead, especially in ASCII text. ) If f ( x ) : = x, then f ′ ( x ) = 2x.  ̇ ( displaystyle ( dot ( , ) ) ) dot ( , ) derivative... dot ; time derivative of calculus x  ̇ ( displaystyle ( dot ( x ) ) ) means the derivative of x with respect to time. That is x  ̇ ( t ) = ∂ ∂ t x ( t ) ( displaystyle ( dot ( x ) ) ( t ) = ( \frac ( partial ) ( partial t ) ) x ( t ) ). If x ( t ) : = t, then x  ̇ ( t ) = 2 t ( displaystyle ( dot ( x ) ) ( t ) = 2t ). Symbols based on Latin letters Symbol in HTML Symbol in TeX Name Explanation Examples Read as Category ∀ ∀ ( displaystyle \forall ) \forall universal quantification for all ; for any ; for each ; for every predicate logic ∀ x, P ( x ) means P ( x ) is true for all x. ∀ n ∈ N, n ≥ n. B B ( displaystyle mathbb ( B ) ) mathbb ( B ) B ( displaystyle mathbf ( B ) ) mathbf ( B ) boolean domain B ; the ( set of ) boolean values ; the ( set of ) truth values ; set theory, boolean algebra B means either ( 0, 1 ), ( false, true ), ( F, T ), or ( ⊥, ⊤ ) ( displaystyle left ( \bot, \top \right ) ). ( ¬ False ) ∈ B C C ( displaystyle mathbb ( C ) ) mathbb ( C ) C ( displaystyle mathbf ( C ) ) mathbf ( C ) complex numbers C ; the ( set of ) complex numbers numbers C means ( a + b i : a, b ∈ R ). i = √ − 1 ∈ C c c ( displaystyle ( mathfrak ( c ) ) ) mathfrak c cardinality of the continuum cardinality of the continuum ; c ; cardinality of the real numbers set theory The cardinality of R ( displaystyle mathbb ( R ) ) is denoted by R ( displaystyle mathbb ( R ) ) or by the symbol c ( displaystyle ( mathfrak ( c ) ) ) ( a lowercase Fraktur letter C ). c = ב 1 ( displaystyle ( mathfrak ( c ) ) = ( \beth ) _ ( 1 ) ) ∂ ∂ ( displaystyle partial ) partial partial derivative partial ; d calculus ∂ f / ∂ x means the partial derivative of f with respect to x, where f is a function on ( x,..., x ). If f ( x, y ) : = x y, then ∂ f / ∂ x = 2xy, boundary boundary of topology ∂ M means the boundary of M ∂ ( x : x ≤ 2 ) = ( x : x = 2 ) degree of a polynomial degree of algebra ∂ f means the degree of the polynomial f. ( This may also be written deg f. ) ∂ ( x − 1 ) = 2 E E ( displaystyle mathbb ( E ) ) mathbb E E ( displaystyle mathrm ( E ) ) mathrm ( E ) expected value expected value probability theory the value of a random variable one would `` expect '' to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained E ( X ) = x 1 p 1 + x 2 p 2 + ⋯ + x k p k p 1 + p 2 + ⋯ + p k ( displaystyle mathbb ( E ) ( X ) = ( \frac ( x_ ( 1 ) p_ ( 1 ) + x_ ( 2 ) p_ ( 2 ) + dotsb + x_ ( k ) p_ ( k ) ) ( p_ ( 1 ) + p_ ( 2 ) + dotsb + p_ ( k ) ) ) ) ∃ ∃ ( displaystyle exists ) exists < / math > < br / > existential quantification there exists ; there is ; there are predicate logic ∃ x : P ( x ) means there is at least one x such that P ( x ) is true. ∃ n ∈ N : n is even. ∃ ! ∃ ! ( displaystyle exists ! ) exists ! uniqueness quantification there exists exactly one predicate logic ∃ ! x : P ( x ) means there is exactly one x such that P ( x ) is true. ∃ ! n ∈ N : n + 5 = 2n. ∈ ∉ ∈ ( displaystyle in ) in ∉ ( displaystyle \notin ) \notin set membership is an element of ; is not an element of everywhere, set theory a ∈ S means a is an element of the set S ; a ∉ S means a is not an element of S. ( 1 / 2 ) ∈ N 2 ∉ N ∌ ∌ ( displaystyle \not \ni ) \not \ni set membership does not contain as an element set theory S ∌ e means the same thing as e ∉ S, where S is a set and e is not an element of S. ∋ ∋ ( displaystyle \ni ) \ni such that symbol such that mathematical logic often abbreviated as `` s.t. '' ; : and are also used to abbreviate `` such that ''. The use of ∋ goes back to early mathematical logic and its usage in this sense is declining. The symbol ∍ ( displaystyle \backepsilon ) ( `` back epsilon '' ) is sometimes specifically used for `` such that '' to avoid confusion with set membership. Choose x ( displaystyle x ) ∋ 2 x ( displaystyle x ) and 3 x ( displaystyle x ). ( Here is used in the sense of `` divides ''. ) set membership contains as an element set theory S ∋ e means the same thing as e ∈ S, where S is a set and e is an element of S. H H ( displaystyle mathbb ( H ) ) mathbb ( H ) H ( displaystyle mathbf ( H ) ) mathbf ( H ) quaternions or Hamiltonian quaternions H ; the ( set of ) quaternions numbers H means ( a + b i + c j + d k : a, b, c, d ∈ R ). N N ( displaystyle mathbb ( N ) ) mathbb ( N ) N ( displaystyle mathbf ( N ) ) mathbf ( N ) natural numbers the ( set of ) natural numbers numbers N means either ( 0, 1, 2, 3,... ) or ( 1, 2, 3,... ). The choice depends on the area of mathematics being studied ; e.g. number theorists prefer the latter ; analysts, set theorists and computer scientists prefer the former. To avoid confusion, always check an author 's definition of N. Set theorists often use the notation ω ( for least infinite ordinal ) to denote the set of natural numbers ( including zero ), along with the standard ordering relation ≤. N = ( a : a ∈ Z ) or N = ( a > 0 : a ∈ Z ) ○ ∘ ( displaystyle circ ) circ Hadamard product entrywise product linear algebra For two matrices ( or vectors ) of the same dimensions A, B ∈ R m × n ( displaystyle A, B in ( mathbb ( R ) ) ^ ( m \times n ) ) the Hadamard product is a matrix of the same dimensions A ∘ B ∈ R m × n ( displaystyle A circ B in ( mathbb ( R ) ) ^ ( m \times n ) ) with elements given by ( A ∘ B ) i, j = ( A ) i, j ⋅ ( B ) i, j ( displaystyle ( A circ B ) _ ( i, j ) = ( A ) _ ( i, j ) cdot ( B ) _ ( i, j ) ). ( 1 2 2 4 ) ∘ ( 1 2 0 0 ) = ( 1 4 0 0 ) ( displaystyle ( \begin ( bmatrix ) 1&2 2&4 end ( bmatrix ) ) circ ( \begin ( bmatrix ) 1&2 0&0 end ( bmatrix ) ) = ( \begin ( bmatrix ) 1&4 0&0 end ( bmatrix ) ) ) ∘ ∘ ( displaystyle circ ) circ function composition composed with set theory f ∘ g is the function such that ( f ∘ g ) ( x ) = f ( g ( x ) ). if f ( x ) : = 2x, and g ( x ) : = x + 3, then ( f ∘ g ) ( x ) = 2 ( x + 3 ). O O ( displaystyle O ) O Big O notation big - oh of Computational complexity theory The Big O notation describes the limiting behavior of a function, when the argument tends towards a particular value or infinity. If f ( x ) = 6x − 2x + 5 and g ( x ) = x, then f ( x ) = O ( g ( x ) ) as x → ∞ ( displaystyle f ( x ) = O ( g ( x ) ) ( mbox ( as ) ) x \to infty , ) ∅ ( ) ∅ ( displaystyle emptyset ) empty ∅ ( displaystyle \varnothing ) \varnothing ( ) ( displaystyle ( ) ) ( ) empty set the empty set null set set theory ∅ means the set with no elements. ( ) means the same. ( n ∈ N : 1 < n < 4 ) = ∅ P P ( displaystyle mathbb ( P ) ) mathbb ( P ) P ( displaystyle mathbf ( P ) ) mathbf ( P ) set of primes P ; the set of prime numbers arithmetic P is often used to denote the set of prime numbers. 2 ∈ P, 3 ∈ P, 8 ∉ P ( displaystyle 2 in mathbb ( P ), 3 in mathbb ( P ), 8 \notin mathbb ( P ) ) projective space P ; the projective space ; the projective line ; the projective plane topology P means a space with a point at infinity. P 1 ( displaystyle mathbb ( P ) ^ ( 1 ) ), P 2 ( displaystyle mathbb ( P ) ^ ( 2 ) ) probability the probability of probability theory P ( X ) means the probability of the event X occurring. This may also be written as P ( X ), Pr ( X ), P ( X ) or Pr ( X ). If a fair coin is flipped, P ( Heads ) = P ( Tails ) = 0.5. Power set the Power set of Powerset Given a set S, the power set of S is the set of all subsets of the set S. The power set of S0 is denoted by P ( S ). The power set P ( ( 0, 1, 2 ) ) is the set of all subsets of ( 0, 1, 2 ). Hence, P ( ( 0, 1, 2 ) ) = ( ∅, ( 0 ), ( 1 ), ( 2 ), ( 0, 1 ), ( 0, 2 ), ( 1, 2 ), ( 0, 1, 2 ) ). Q Q Q ( displaystyle mathbb ( Q ) ) mathbb ( Q ) Q ( displaystyle mathbf ( Q ) ) mathbf ( Q ) rational numbers Q ; the ( set of ) rational numbers ; the rationals numbers Q means ( p / q : p ∈ Z, q ∈ N ). 3.14000... ∈ Q π ∉ Q R R ( displaystyle mathbb ( R ) ) mathbb ( R ) R ( displaystyle mathbf ( R ) ) mathbf ( R ) real numbers R ; the ( set of ) real numbers ; the reals numbers R means the set of real numbers. π ∈ R √ ( − 1 ) ∉ R † ( displaystyle ( ) ^ ( dagger ) ) ( ) ^ dagger conjugate transpose conjugate transpose ; adjoint ; Hermitian adjoint / conjugate / transpose / dagger matrix operations A means the transpose of the complex conjugate of A. This may also be written A, A, A, A or A. If A = ( a ) then A = ( a ). T ( displaystyle ( ) ^ ( mathsf ( T ) ) ) ( ) ^ ( mathsf ( T ) ) transpose transpose matrix operations A means A, but with its rows swapped for columns. This may also be written A ′, A or A. If A = ( a ) then A = ( a ). ⊤ ⊤ ( displaystyle \top ) \top top element the top element lattice theory ⊤ means the largest element of a lattice. ∀ x : x ∨ ⊤ = ⊤ top type the top type ; top type theory ⊤ means the top or universal type ; every type in the type system of interest is a subtype of top. ∀ types T, T < : ⊤ ⊥ ⊥ ( displaystyle \bot ) \bot perpendicular is perpendicular to geometry x ⊥ y means x is perpendicular to y ; or more generally x is orthogonal to y. If l ⊥ m and m ⊥ n in the plane, then l n. orthogonal complement orthogonal / perpendicular complement of ; perp linear algebra W means the orthogonal complement of W ( where W is a subspace of the inner product space V ), the set of all vectors in V orthogonal to every vector in W. Within R 3 ( displaystyle mathbb ( R ) ^ ( 3 ) ), ( R 2 ) ⊥ ≅ R ( displaystyle ( mathbb ( R ) ^ ( 2 ) ) ^ ( perp ) cong mathbb ( R ) ). coprime is coprime to number theory x ⊥ y means x has no factor greater than 1 in common with y. 34 ⊥ 55 independent is independent of probability A ⊥ B means A is an event whose probability is independent of event B. The double perpendicular symbol ( ⊥ ⊥ ( displaystyle perp ! ! ! perp ) ) is also commonly used for the purpose of denoting this, for instance : A ⊥ ⊥ B ( displaystyle A perp ! ! ! perp B ) ( In LaTeX, the command is : `` A perp ! ! ! perp B ''. ) If A ⊥ B, then P ( A B ) = P ( A ). bottom element the bottom element lattice theory ⊥ means the smallest element of a lattice. ∀ x : x ∧ ⊥ = ⊥ bottom type the bottom type ; bot type theory ⊥ means the bottom type ( a.k.a. the zero type or empty type ) ; bottom is the subtype of every type in the type system. ∀ types T, ⊥ < : T comparability is comparable to order theory x ⊥ y means that x is comparable to y. ( e, π ) ⊥ ( 1, 2, e, 3, π ) under set containment. U U U ( displaystyle mathbb ( U ) ) mathbb ( U ) U ( displaystyle mathbf ( U ) ) mathbf ( U ) all numbers being considered U ; the universal set ; the set of all numbers ; all numbers considered set theory U means `` the set of all elements being considered. '' It may represent all numbers both real and complex, or any subset of these -- hence the term `` universal ''. U = ( R, C ) includes all numbers. If instead, U = ( Z, C ), then π ∉ U. ∪ ∪ ( displaystyle cup ) cup set - theoretic union the union of... or... ; union set theory A ∪ B means the set of those elements which are either in A, or in B, or in both. A ⊆ B ⇔ ( A ∪ B) = B ∩ ∩ ( displaystyle cap ) cap set - theoretic intersection intersected with ; intersect set theory A ∩ B means the set that contains all those elements that A and B have in common. ( x ∈ R : x = 1 ) ∩ N = ( 1 ) ∨ ∨ ( displaystyle lor ) or logical disjunction or join in a lattice or ; max ; join propositional logic, lattice theory The 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. ∧ ∧ ( displaystyle land ) and logical conjunction or meet in a lattice and ; min ; meet propositional logic, lattice theory The 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 product wedge product ; exterior product exterior algebra u ∧ v means the wedge product of any multivectors u and v. In three - dimensional Euclidean space the wedge product and the cross product of two vectors are each other 's Hodge dual. u ∧ v = ∗ ( u × v ) if u, v ∈ R 3 ( displaystyle u wedge v = * ( u \times v ) ( \text ( if ) ) u, v in mathbb ( R ) ^ ( 3 ) ) × × ( displaystyle \times ) \times multiplication times ; multiplied by arithmetic 3 × 4 means the multiplication of 3 by 4. ( The symbol * is generally used in programming languages, where ease of typing and use of ASCII text is preferred. ) 7 × 8 = 56 Cartesian product the Cartesian product of... and... ; the direct product of... and... set theory 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 ) ) cross product cross linear algebra u × v means the cross product of vectors u and v ( 1, 2, 5 ) × ( 3, 4, − 1 ) = ( − 22, 16, − 2 ) group of units the group of units of ring theory R 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, or U ( R ). ( Z / 5 Z ) × = ( ( 1 ), ( 2 ), ( 3 ), ( 4 ) ) ≅ C 4 ( displaystyle ( \begin ( aligned ) ( mathbb ( Z ) / 5 mathbb ( Z ) ) ^ ( \times ) & = ( ( 1 ), ( 2 ), ( 3 ), ( 4 ) ) & cong mathrm ( C ) _ ( 4 ) end ( aligned ) ) ) ⊗ ⊗ ( displaystyle otimes ) otimes tensor product, tensor product of modules tensor product of linear algebra V ⊗ U ( displaystyle V otimes U ) means the tensor product of V and U. V ⊗ R U ( displaystyle V otimes _ ( R ) U ) means the tensor product of modules V and U over the ring R. ( 1, 2, 3, 4 ) ⊗ ( 1, 1, 2 ) = ( ( 1, 1, 2 ), ( 2, 2, 4 ), ( 3, 3, 6 ), ( 4, 4, 8 ) ) ⋉ ⋊ ⋉ ( displaystyle ltimes ) ltimes ⋊ ( displaystyle \rtimes ) \rtimes semidirect product the semidirect product of group theory N ⋊ H is the semidirect product of N ( a normal subgroup ) and H ( a subgroup ), with respect to φ. Also, if G = N ⋊ H, then G is said to split over N. ( ⋊ may also be written the other way round, as ⋉, or as ×. ) D 2 n ≅ C n ⋊ C 2 ( displaystyle D_ ( 2n ) cong mathrm ( C ) _ ( n ) \rtimes mathrm ( C ) _ ( 2 ) ) semijoin the semijoin of relational algebra R ⋉ S is the semijoin of the relations R and S, the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names. R ⋉ ( displaystyle ltimes ) S = Π ( displaystyle Pi ) ( R ⋈ ( displaystyle \bowtie ) S ) ⋈ ⋈ ( displaystyle \bowtie ) \bowtie natural join the natural join of relational algebra R ⋈ 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. Z Z Z ( displaystyle mathbb ( Z ) ) mathbb ( Z ) Z ( displaystyle mathbf ( Z ) ) mathbf ( Z ) integers the ( set of ) integers numbers Z means (..., − 3, − 2, − 1, 0, 1, 2, 3,... ). Z or Z means ( 1, 2, 3,... ). Z means ( 0, 1, 2, 3,... ). Z is used by some authors to mean ( 0, 1, 2, 3,... ) and others to mean (... - 2, - 1, 1, 2, 3,... ). Z = ( p, − p : p ∈ N ∪ ( 0 ) ​ ) Z Z Z Z Z n ( displaystyle mathbb ( Z ) _ ( n ) ) mathbb ( Z ) _n Z p ( displaystyle mathbb ( Z ) _ ( p ) ) mathbb ( Z ) _p Z n ( displaystyle mathbf ( Z ) _ ( n ) ) mathbf ( Z ) _n Z p ( displaystyle mathbf ( Z ) _ ( p ) ) integers mod n the ( set of ) integers modulo n numbers Z means ( ( 0 ), ( 1 ), ( 2 ),... ( n − 1 ) ) with addition and multiplication modulo n. Note that any letter may be used instead of n, such as p. To avoid confusion with p - adic numbers, use Z / pZ or Z / ( p ) instead. Z = ( ( 0 ), ( 1 ), ( 2 ) ) p - adic integers the ( set of ) p - adic integers numbers Note that any letter may be used instead of p, such as n or l. Symbols based on Hebrew or Greek letters Symbol in HTML Symbol in TeX Name Explanation Examples Read as Category א א ( displaystyle aleph ) aleph aleph number aleph set theory א represents an infinite cardinality ( specifically, the α - th one, where α is an ordinal ). N = א, which is called aleph - null. ב ב ( displaystyle \beth ) \beth beth number beth set theory ב represents an infinite cardinality ( similar to א, but ב does not necessarily index all of the numbers indexed by א. ). ב 1 = P ( N ) = 2 א 0. ( displaystyle \beth _ ( 1 ) = P ( mathbb ( N ) ) = 2 ^ ( aleph _ ( 0 ) ). ) δ δ ( displaystyle delta ) delta Dirac delta function Dirac delta of hyperfunction δ ( x ) = ( ∞, x = 0 0, x ≠ 0 ( displaystyle delta ( x ) = ( \begin ( cases ) infty, &x = 0 0, &x \neq 0 end ( cases ) ) ) δ ( x ) Kronecker delta Kronecker delta of hyperfunction δ i j = ( 1, i = j 0, i ≠ j ( displaystyle delta _ ( ij ) = ( \begin ( cases ) 1, &i = j 0, &i \neq j end ( cases ) ) ) δ Functional derivative Functional derivative of Differential operators ⟨ δ F ( φ ( x ) ) δ φ ( x ), f ( x ) ⟩ = ∫ δ F ( φ ( x ) ) δ φ ( x ′ ) f ( x ′ ) d x ′ = lim ε → 0 F ( φ ( x ) + ε f ( x ) ) − F ( φ ( x ) ) ε = d d ε F ( φ + ε f ) ε = 0. ( displaystyle ( \begin ( aligned ) left langle ( \frac ( delta F ( \varphi ( x ) ) ) ( delta \varphi ( x ) ) ), f ( x ) \right \rangle & = int ( \frac ( delta F ( \varphi ( x ) ) ) ( delta \varphi ( x ' ) ) ) f ( x ' ) dx ' & = lim _ ( \varepsilon \to 0 ) ( \frac ( F ( \varphi ( x ) + \varepsilon f ( x ) ) - F ( \varphi ( x ) ) ) ( \varepsilon ) ) & = left. ( \frac ( d ) ( d epsilon ) ) F ( \varphi + epsilon f ) \right _ ( epsilon = 0 ). end ( aligned ) ) ) δ V ( r ) δ ρ ( r ′ ) = 1 4 π ε 0 r − r ′ ( displaystyle ( \frac ( delta V ( r ) ) ( delta \rho ( r ' ) ) ) = ( \frac ( 1 ) ( 4 pi epsilon _ ( 0 ) r-r ' ) ) ) ∆ ⊖ ⊕ △ ( displaystyle \vartriangle ) \vartriangle ⊖ ( displaystyle ominus ) ominus ⊕ ( displaystyle oplus ) oplus symmetric difference symmetric difference set theory A ∆ B ( or A ⊖ B) means the set of elements in exactly one of A or B. ( Not to be confused with delta, Δ, described below. ) ( 1, 5, 6, 8 ) ∆ ( 2, 5, 8 ) = ( 1, 2, 6 ) ( 3, 4, 5, 6 ) ⊖ ( 1, 2, 5, 6 ) = ( 1, 2, 3, 4 ) Δ Δ ( displaystyle Delta ) Delta delta delta ; change in calculus Δx means a ( non-infinitesimal ) change in x. ( If the change becomes infinitesimal, δ and even d are used instead. Not to be confused with the symmetric difference, written ∆, above. ) Δ y Δ x ( displaystyle ( \tfrac ( Delta y ) ( Delta x ) ) ) is the gradient of a straight line. Laplacian Laplace operator vector calculus The Laplace operator is a second order differential operator in n - dimensional Euclidean space If ƒ is a twice - differentiable real - valued function, then the Laplacian of ƒ is defined by Δ f = ∇ 2 f = ∇ ⋅ ∇ f ( displaystyle Delta f = \nabla ^ ( 2 ) f = \nabla cdot \nabla f ) ∇ ∇ ( displaystyle \nabla ) \nabla gradient del ; nabla ; gradient of vector calculus ∇ f ( x,..., x ) is the vector of partial derivatives ( ∂ f / ∂ x,..., ∂ f / ∂ x ). If f ( x, y, z ) : = 3xy + z2, then ∇ f = ( 3y, 3x, 2z ) divergence del dot ; divergence of vector calculus ∇ ⋅ v → = ∂ v x ∂ x + ∂ v y ∂ y + ∂ v z ∂ z ( displaystyle \nabla cdot ( \vec ( v ) ) = ( partial v_ ( x ) over partial x ) + ( partial v_ ( y ) over partial y ) + ( partial v_ ( z ) over partial z ) ) If v → : = 3 x y i + y 2 z j + 5 k ( displaystyle ( \vec ( v ) ) : = 3xy mathbf ( i ) + y ^ ( 2 ) z mathbf ( j ) + 5 mathbf ( k ) ), then ∇ ⋅ v → = 3 y + 2 y z ( displaystyle \nabla cdot ( \vec ( v ) ) = 3y + 2yz ). curl curl of vector calculus ∇ × v → = ( ∂ v z ∂ y − ∂ v y ∂ z ) i ( displaystyle \nabla \times ( \vec ( v ) ) = left ( ( partial v_ ( z ) over partial y ) - ( partial v_ ( y ) over partial z ) \right ) mathbf ( i ) ) + ( ∂ v x ∂ z − ∂ v z ∂ x ) j + ( ∂ v y ∂ x − ∂ v x ∂ y ) k ( displaystyle + left ( ( partial v_ ( x ) over partial z ) - ( partial v_ ( z ) over partial x ) \right ) mathbf ( j ) + left ( ( partial v_ ( y ) over partial x ) - ( partial v_ ( x ) over partial y ) \right ) mathbf ( k ) ) If v → : = 3 x y i + y 2 z j + 5 k ( displaystyle ( \vec ( v ) ) : = 3xy mathbf ( i ) + y ^ ( 2 ) z mathbf ( j ) + 5 mathbf ( k ) ), then ∇ × v → = − y 2 i − 3 x k ( displaystyle \nabla \times ( \vec ( v ) ) = - y ^ ( 2 ) mathbf ( i ) - 3x mathbf ( k ) ). π π ( displaystyle pi ) pi Pi pi ; 3.1415926... ; ≈ 355 ÷ 113 mathematical constant Used in various formulas involving circles ; π is equivalent to the amount of area a circle would take up in a square of equal width with an area of 4 square units, roughly 3.14159. It is also the ratio of the circumference to the diameter of a circle. A = π R = 314.16 → R = 10 projection Projection of relational algebra π a 1,..., a n ( R ) ( displaystyle pi _ ( a_ ( 1 ), ldots, a_ ( n ) ) ( R ) ) restricts R ( displaystyle R ) to the ( a 1,..., a n ) ( displaystyle ( a_ ( 1 ), ldots, a_ ( n ) ) ) attribute set. π Age, Weight ( Person ) ( displaystyle pi _ ( \text ( Age, Weight ) ) ( ( \text ( Person ) ) ) ) Homotopy group the nth Homotopy group of Homotopy theory π n ( X ) ( displaystyle pi _ ( n ) ( X ) ) consists of homotopy equivalence classes of base point preserving maps from an n - dimensional sphere ( with base point ) into the pointed space X. π i ( S 4 ) = π i ( S 7 ) ⊕ π i − 1 ( S 3 ) ( displaystyle pi _ ( i ) ( S ^ ( 4 ) ) = pi _ ( i ) ( S ^ ( 7 ) ) oplus pi _ ( i - 1 ) ( S ^ ( 3 ) ) ) ∏ ∏ ( displaystyle prod ) prod product product over... from... to... of arithmetic ∏ k = 1 n a k ( displaystyle prod _ ( k = 1 ) ^ ( n ) a_ ( k ) ) means a 1 a 2... a n ( displaystyle a_ ( 1 ) a_ ( 2 ) dots a_ ( n ) ). ∏ k = 1 4 ( k + 2 ) = ( 1 + 2 ) ( 2 + 2 ) ( 3 + 2 ) ( 4 + 2 ) = 3 × 4 × 5 × 6 = 360 ( displaystyle prod _ ( k = 1 ) ^ ( 4 ) ( k + 2 ) = ( 1 + 2 ) ( 2 + 2 ) ( 3 + 2 ) ( 4 + 2 ) = 3 \times 4 \times 5 \times 6 = 360 ) Cartesian product the Cartesian product of ; the direct product of set theory ∏ i = 0 n Y i ( displaystyle prod _ ( i = 0 ) ^ ( n ) ( Y_ ( i ) ) ) means the set of all ( n + 1 ) - tuples ( y,..., y ). ∏ n = 1 3 R = R × R × R = R 3 ( displaystyle prod _ ( n = 1 ) ^ ( 3 ) ( mathbb ( R ) ) = mathbb ( R ) \times mathbb ( R ) \times mathbb ( R ) = mathbb ( R ) ^ ( 3 ) ) ∐ ∐ ( displaystyle coprod ) coprod coproduct coproduct over... from... to... of category theory A general construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the `` least specific '' object to which each object in the family admits a morphism. σ σ ( displaystyle sigma ) sigma selection Selection of relational algebra The selection σ a θ b ( R ) ( displaystyle sigma _ ( a \theta b ) ( R ) ) selects all those tuples in R ( displaystyle R ) for which θ ( displaystyle \theta ) holds between the a ( displaystyle a ) and the b ( displaystyle b ) attribute. The selection σ a θ v ( R ) ( displaystyle sigma _ ( a \theta v ) ( R ) ) selects all those tuples in R ( displaystyle R ) for which θ ( displaystyle \theta ) holds between the a ( displaystyle a ) attribute and the value v ( displaystyle v ). σ A g e ≥ 34 ( P e r s o n ) ( displaystyle sigma _ ( mathrm ( Age ) geq 34 ) ( mathrm ( Person ) ) ) σ A g e = W e i g h t ( P e r s o n ) ( displaystyle sigma _ ( mathrm ( Age ) = mathrm ( Weight ) ) ( mathrm ( Person ) ) ) ∑ ∑ ( displaystyle sum ) sum summation sum over... from... to... of arithmetic ∑ k = 1 n a k ( displaystyle sum _ ( k = 1 ) ^ ( n ) ( a_ ( k ) ) ) means a 1 + a 2 + ⋯ + a n ( displaystyle a_ ( 1 ) + a_ ( 2 ) + cdots + a_ ( n ) ). ∑ k = 1 4 k 2 = 1 2 + 2 2 + 3 2 + 4 2 = 1 + 4 + 9 + 16 = 30 ( displaystyle sum _ ( k = 1 ) ^ ( 4 ) ( k ^ ( 2 ) ) = 1 ^ ( 2 ) + 2 ^ ( 2 ) + 3 ^ ( 2 ) + 4 ^ ( 2 ) = 1 + 4 + 9 + 16 = 30 ) Variations In mathematics written in Persian or Arabic, some symbols may be reversed to make right - to - left writing and reading easier. See also Greek letters used in mathematics, science, and engineering Diacritic ISO 31 - 11 ( Mathematical signs and symbols for use in physical sciences and technology ) Latin letters used in mathematics List of mathematical abbreviations List of mathematical symbols by subject Mathematical Alphanumeric Symbols ( Unicode block ) Mathematical constants and functions Mathematical notation Mathematical operators and symbols in Unicode Notation in probability and statistics Physical constants Table of logic symbols Table of mathematical symbols by introduction date Typographical conventions in mathematical formulae References ^ `` LaTeX / Mathematics ''. Wikibooks. Retrieved 18 November 2017. ^ `` The Comprehensive LATEX Symbol List '' ( PDF ). p. 15. Retrieved 16 November 2017. Because of the lack of notational consensus, it is probably better to spell out `` Contradiction ! '' than to use asymbol for this purpose. ^ Cook, John. `` Unicode / LaTeX conversion ''. John Cook Consulting. Retrieved 18 November 2017. ^ `` LaTeX / Special Characters ''. Wikibooks. Retrieved 18 November 2017. ^ `` \unicode - Tex Command ''. TutorialsBay. Retrieved 18 November 2017. ^ `` Unicode characters in pdflatex output using hexcode without UTF - 8 input ''. Tex Stack Exchange. Retrieved 18 November 2017. ^ `` fontenc vs inputenc ''. TeX Stack Exchange. Retrieved 18 November 2017. ^ `` pdflatex crashes when Latex code includes \unicode ( f818 ) and \unicode ( f817 ) and how to handle it ''. TeX Stack Exchange. Retrieved 18 November 2017. ^ `` Math is Fun website ''. ^ Rónyai, Lajos ( 1998 ), Algoritmusok ( Algorithms ), TYPOTEX, ISBN 963 - 9132 - 16 - 0 ^ Deb, K. ; Pratap, A. ; Agarwal, S. ; Meyarivan, T. ( 2002 ). `` A fast and elitist multiobjective genetic algorithm : NSGA - II ''. IEEE Transactions on Evolutionary Computation. 6 ( 2 ) : 182. doi : 10.1109 / 4235.996017. ^ Copi, Irving M. ; Cohen, Carl ( 1990 ) ( 1953 ), `` Chapter 8.3 : Conditional Statements and Material Implication '', Introduction to Logic ( 8th ed. ), New York : Macmillan Publishers ( United States ), pp. 268 -- 269, ISBN 0 - 02 - 325035 - 6, LCCN 89037742 ^ Goldrei, Derek ( 1996 ), Classic Set Theory, London : Chapman and Hall, p. 4, ISBN 0 - 412 - 60610 - 0 ^ Nielsen, Michael A ; Chuang, Isaac L ( 2000 ), Quantum Computation and Quantum Information, New York : Cambridge University Press, p. 66, ISBN 0 - 521 - 63503 - 9, OCLC 43641333 ^ Goldrei, Derek ( 1996 ), Classic Set Theory, London : Chapman and Hall, p. 3, ISBN 0 - 412 - 60610 - 0 ^ `` Expectation Value ''. http://mathworld.wolfram.com. Retrieved 2017 - 12 - 02. External link in website = ( help ) ^ Nielsen, Michael A ; Chuang, Isaac L ( 2000 ), Quantum Computation and Quantum Information, New York : Cambridge University Press, p. 62, ISBN 0 - 521 - 63503 - 9, OCLC 43641333 ^ Berman, Kenneth A ; Paul, Jerome L. ( 2005 ), Algorithms : Sequential, Parallel, and Distributed, Boston : Course Technology, p. 822, ISBN 0 - 534 - 42057 - 5 ^ `` Parallel Symbol in TeX ''. Google Groups. Retrieved 16 November 2017. ^ `` Math symbols defined by LaTeX package `` stmaryrd '' '' ( PDF ). Retrieved 16 November 2017. ^ `` Answer to Is there a `` contradiction '' symbol in some font, somewhere ? ``. TeX Stack Exchange. Retrieved 16 November 2017. ^ Goldrei, Derek ( 1996 ), Classic Set Theory, London : Chapman and Hall, p. 5, ISBN 0 - 412 - 60610 - 0 ^ Nielsen, Michael A ; Chuang, Isaac L ( 2000 ), Quantum Computation and Quantum Information, New York : Cambridge University Press, pp. 69 -- 70, ISBN 0 - 521 - 63503 - 9, OCLC 43641333 ^ Nielsen, Michael A ; Chuang, Isaac L ( 2000 ), Quantum Computation and Quantum Information, New York : Cambridge University Press, pp. 71 -- 72, ISBN 0 - 521 - 63503 - 9, OCLC 43641333 ^ Z ^ * from Wolfram MathWorld ^ LK Turner, FJ BUdden, D Knighton, `` Advanced Mathematics '', Book 2, Longman 1975. ^ M. Benatia, A. Lazrik, and K. Sami, `` Arabic mathematical symbols in Unicode '', 27th Internationalization and Unicode Conference, 2005. External links The complete set of mathematics Unicode characters Jeff Miller : Earliest Uses of Various Mathematical Symbols Numericana : Scientific Symbols and Icons GIF and PNG Images for Math Symbols Mathematical Symbols in Unicode Using Greek and special characters from Symbol font in HTML DeTeXify handwritten symbol recognition -- doodle a symbol in the box, and the program will tell you what its name is Handbook for Spoken Mathematics -- pronunciation guide to many commonly used symbols Some Unicode charts of mathematical operators and symbols : Index of Unicode symbols Range 2100 -- 214F : Unicode Letterlike Symbols Range 2190 -- 21FF : Unicode Arrows Range 2200 -- 22FF : Unicode Mathematical Operators Range 27C0 -- 27EF : Unicode Miscellaneous Mathematical Symbols -- A Range 2980 -- 29FF : Unicode Miscellaneous Mathematical Symbols -- B Range 2A00 -- 2AFF : Unicode Supplementary Mathematical Operators Some Unicode cross-references : Short list of commonly used LaTeX symbols and Comprehensive LaTeX Symbol List MathML Characters - sorts out Unicode, HTML and MathML / TeX names on one page Unicode values and MathML names Unicode values and Postscript names from the source code for Ghostscript Retrieved from `` https://en.wikipedia.org/w/index.php?title=List_of_mathematical_symbols&oldid=814439780 '' Categories : Mathematical notation Mathematics - related lists Mathematical symbols Mathematical tables Mathematical logic Lists of symbols Hidden categories : CS1 errors : external links Incomplete lists from March 2010 Talk Contents About Wikipedia አማርኛ বাংলা Български Bosanski Català Čeština Cymraeg Deutsch Español Euskara فارسی Français Galego 한국어 Հայերեն हिन्दी Bahasa Indonesia Italiano עברית Қазақша Lumbaart Magyar മലയാളം Nederlands 日本 語 Norsk Norsk nynorsk پښتو Polski Português Română Русский Simple English Slovenščina Soomaaliga Basa Sunda Suomi Svenska Türkçe Українська اردو Tiếng Việt Võro 文言 粵語 中文 Edit links This page was last edited on 8 December 2017, at 21 : 03. 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Question: what does the e symbol mean in maths
Answer: summation