Operators

Basic Mathematical Operators

form
full name
alias
×\[Times]Esc*Esc
÷\[Divide]EscdivEsc
\[Sqrt]EscsqrtEsc
form
full name
alias
\[Cross]EsccrossEsc
±\[PlusMinus]Esc+-Esc
\[MinusPlus]Esc-+Esc

Some operators used in basic arithmetic and algebra.

Note that the for [Cross] is distinguished by being drawn slightly smaller than the for [Times].

x×yTimes[x,y]multiplication
x÷yDivide[x,y]division
xSqrt[x]square root
xyCross[x,y]vector cross product
±xPlusMinus[x](no builtin meaning)
x±yPlusMinus[x,y](no builtin meaning)
xMinusPlus[x](no builtin meaning)
xyMinusPlus[x,y](no builtin meaning)

Interpretation of some operators in basic arithmetic and algebra.

Operators in Calculus and Algebra

form
full name
alias
\[Del]EscdelEsc
\[PartialD]EscpdEsc
\[DifferentialD]EscddEsc
\[Sum]EscsumEsc
\[Product]EscprodEsc
form
full name
alias
\[Integral]EscintEsc
\[ContourIntegral]EsccintEsc
\[DoubleContourIntegral]
\[CounterClockwiseContourIntegral]EsccccintEsc
\[ClockwiseContourIntegral]EscccintEsc

Operators used in calculus.

form
full name
aliases
\[Conjugate]EsccoEsc, EscconjEsc
\[Transpose]EsctrEsc
form
full name
alias
\[ConjugateTranspose]EscctEsc
\[HermitianConjugate]EschcEsc

Operators for complex numbers and matrices.

Logical and Other Connectives

form
full name
aliases
\[And]Esc, EscandEsc
\[Or]Esc, EscorEsc
¬\[Not]Esc, EscnotEsc
\[Element]EscelEsc
\[ForAll]EscfaEsc
\[Exists]EscexEsc
\[NotExists]Esc!exEsc
\[Xor]EscxorEsc
\[Nand]EscnandEsc
\[Nor]EscnorEsc
form
full name
alias
\[Implies]Esc=>Esc
\[RoundImplies]
\[Therefore]EsctfEsc
\[Because]
\[RightTee]
\[LeftTee]
\[DoubleRightTee]
\[DoubleLeftTee]
\[SuchThat]EscstEsc
\[VerticalSeparator]Esc|Esc
\[Colon]Esc:Esc

Operators used as logical connectives.

The operators , , and are interpreted as corresponding to the builtin functions And, Or, and Not, and are equivalent to the keyboard operators , , and . The operators , , and correspond to the builtin functions Xor, Nand, and Nor. Note that is a prefix operator.

and are both taken to give the builtin function Implies[x,y]. gives the builtin function Element[x,y].

This is interpreted using the builtin functions And and Implies.
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The Wolfram Language supports most of the standard syntax used in mathematical logic. In the Wolfram Language, however, the variables that appear in the quantifiers , , and must appear as subscripts. If they appeared directly after the quantifier symbols then there could be a conflict with multiplication operations.

and are essentially prefix operators like .
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Operators Used to Represent Actions

form
full name
alias
\[SmallCircle]EscscEsc
\[CirclePlus]Escc+Esc
\[CircleMinus]Escc-Esc
\[CircleTimes]Escc*Esc
\[CircleDot]Escc.Esc
\[Diamond]EscdiaEsc
·\[CenterDot]Esc.Esc
\[Star]EscstarEsc
\[VerticalTilde]
\[Backslash]Esc\Esc
form
full name
alias
\[Wedge]Esc^Esc
\[Vee]EscvEsc
\[Union]EscunEsc
\[UnionPlus]
\[Intersection]EscinterEsc
\[SquareIntersection]
\[SquareUnion]
\[Coproduct]EsccoprodEsc
\[Cap]
\[Cup]
\[Square]EscsqEsc

Operators typically used to represent actions. All the operators except [Square] are infix.

Following the Wolfram Language's usual convention, all the operators in the table are interpreted to give functions whose names are exactly the names of the characters that appear in the operators.

The operators are interpreted as functions with corresponding names.
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All the operators in the table above, except for , are infix, so that they must appear in between their operands.

Bracketing Operators

form
full name
alias
\[LeftFloor]EsclfEsc
\[RightFloor]EscrfEsc
\[LeftCeiling]EsclcEsc
\[RightCeiling]EscrcEsc
\[LeftDoubleBracket]Esc[[Esc
\[RightDoubleBracket]Esc]]Esc
form
full name
alias
\[LeftAngleBracket]Esc<Esc
\[RightAngleBracket]Esc>Esc
\[LeftBracketingBar]Escl|Esc
\[RightBracketingBar]Escr|Esc
\[LeftDoubleBracketingBar]Escl||Esc
\[RightDoubleBracketingBar]Escr||Esc

Characters used as bracketing operators.

xFloor[x]
xCeiling[x]
mi,j,Part[m,i,j,]
x,y,AngleBracket[x,y,]
x,y,BracketingBar[x,y,]
x,y,DoubleBracketingBar[x,y,]

Interpretations of bracketing operators.

Operators Used to Represent Relations

form
full name
alias
\[Equal]Esc==Esc
\[LongEqual]Escl=Esc
\[Congruent]Esc===Esc
\[Tilde]EscEsc
\[TildeTilde]EscEsc
\[TildeEqual]Esc=Esc
\[TildeFullEqual]Esc==Esc
\[EqualTilde]EscEsc
\[HumpEqual]Esch=Esc
\[HumpDownHump]
\[CupCap]
\[DotEqual]
form
full name
alias
\[NotEqual]Esc!=Esc
\[NotCongruent]Esc!===Esc
\[NotTilde]EscEsc
\[NotTildeTilde]EscEsc
\[NotTildeEqual]Esc=Esc
\[NotTildeFullEqual]Esc==Esc
\[NotEqualTilde]EscEsc
\[NotHumpEqual]Esc!h=Esc
\[NotHumpDownHump]
\[NotCupCap]
\[Proportional]EscpropEsc
\[Proportion]

Operators usually used to represent similarity or equivalence.

The special character (or [Equal]) is an alternative input form for .  is used both for input and output.
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Operators usually used for ordering by magnitude.

form
full name
alias
\[Subset]EscsubEsc
\[Superset]EscsupEsc
\[SubsetEqual]Escsub=Esc
\[SupersetEqual]Escsup=Esc
\[Element]EscelEsc
\[ReverseElement]EscmemEsc
form
full name
alias
\[NotSubset]Esc!subEsc
\[NotSuperset]Esc!supEsc
\[NotSubsetEqual]Esc!sub=Esc
\[NotSupersetEqual]Esc!sup=Esc
\[NotElement]Esc!elEsc
\[NotReverseElement]Esc!memEsc

Operators used for relations in sets.

Operators usually used for other kinds of orderings.

form
full name
alias
\[VerticalBar]Esc|Esc
\[DoubleVerticalBar]Esc||Esc
form
full name
alias
\[NotVerticalBar]Esc!|Esc
\[NotDoubleVerticalBar]Esc!||Esc

Relational operators based on vertical bars.

Operators Based on Arrows and Vectors

Operators based on arrows are often used in pure mathematics and elsewhere to represent various kinds of transformations or changes.

is equivalent to .
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form
full name
alias
\[Rule]Esc->Esc
\[RuleDelayed]Esc:>Esc
form
full name
alias
\[Implies]Esc=>Esc
\[RoundImplies]

Arrowlike operators with builtin meanings in the Wolfram Language.

Ordinary arrows.

Vectors and related arrows.

All the arrow and vectorlike operators in the Wolfram Language are infix.
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form
full name
alias
\[RightTee]EscrTEsc
\[LeftTee]EsclTEsc
\[UpTee]EscuTEsc
\[DownTee]EscdTEsc
form
full name
\[DoubleRightTee]
\[DoubleLeftTee]

Tees.