# 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×y Times[x,y] multiplication x÷y Divide[x,y] division √x Sqrt[x] square root xy Cross[x,y] vector cross product ±x PlusMinus[x] (no built‐in meaning) x±y PlusMinus[x,y] (no built‐in meaning) ∓x MinusPlus[x] (no built‐in meaning) x∓y MinusPlus[x,y] (no built‐in 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.

xy and xy are both taken to give the builtin function Implies[x,y]. xy 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] EscEsc  \[LeftBracketingBar] Escl|Esc  \[RightBracketingBar] Escr|Esc  \[LeftDoubleBracketingBar] Escl||Esc  \[RightDoubleBracketingBar] Escr||Esc

Characters used as bracketing operators.

 ⌊x⌋ Floor[x] ⌈x⌉ Ceiling[x] m〚i,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] Esc~Esc ≈ \[TildeTilde] Esc~~Esc ≃ \[TildeEqual] Esc~=Esc ≅ \[TildeFullEqual] Esc~==Esc ≂ \[EqualTilde] Esc=~Esc ≏ \[HumpEqual] Esch=Esc ≎ \[HumpDownHump] ≍ \[CupCap] ≐ \[DotEqual]
 form full name alias ≠ \[NotEqual] Esc!=Esc ≢ \[NotCongruent] Esc!===Esc ≁ \[NotTilde] Esc!~Esc ≉ \[NotTildeTilde] Esc!~~Esc ≄ \[NotTildeEqual] Esc!~=Esc ≇ \[NotTildeFullEqual] Esc!~==Esc  \[NotEqualTilde] Esc!=~Esc  \[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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 form full name alias ≥ \[GreaterEqual] Esc>=Esc ≤ \[LessEqual] Esc<=Esc ⩾ \[GreaterSlantEqual] Esc>/Esc ⩽ \[LessSlantEqual] Esc~Esc ≲ \[LessTilde] Esc<~Esc ≫ \[GreaterGreater] ≪ \[LessLess] ⪢ \[NestedGreaterGreater] ⪡ \[NestedLessLess] ≷ \[GreaterLess] ≶ \[LessGreater] ⋛ \[GreaterEqualLess] ⋚ \[LessEqualGreater]
 form full name alias ≱ \[NotGreaterEqual] Esc!>=Esc ≰ \[NotLessEqual] Esc!<=Esc  \[NotGreaterSlantEqual] Esc!>/Esc  \[NotLessSlantEqual] Esc!~Esc ≴ \[NotLessTilde] Esc!<~Esc  \[NotGreaterGreater]  \[NotLessLess]  \[NotNestedGreaterGreater]  \[NotNestedLessLess] ≹ \[NotGreaterLess] ≸ \[NotLessGreater] ≯ \[NotGreater] Esc!>Esc ≮ \[NotLess] Esc!

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.

 form full name ≻ \[Succeeds] ≺ \[Precedes] ⪰ \[SucceedsEqual] ⪯ \[PrecedesEqual] ≽ \[SucceedsSlantEqual] ≼ \[PrecedesSlantEqual] ≿ \[SucceedsTilde] ≾ \[PrecedesTilde] ⊳ \[RightTriangle] ⊲ \[LeftTriangle] ⊵ \[RightTriangleEqual] ⊴ \[LeftTriangleEqual] ⧐ \[RightTriangleBar] ⧏ \[LeftTriangleBar] ⊐ \[SquareSuperset] ⊏ \[SquareSubset] ⊒ \[SquareSupersetEqual] ⊑ \[SquareSubsetEqual]
 form full name ⊁ \[NotSucceeds] ⊀ \[NotPrecedes]  \[NotSucceedsEqual] ⋨ \[NotPrecedesTilde] ⋡ \[NotSucceedsSlantEqual] ⋠ \[NotPrecedesSlantEqual] ⋩ \[NotSucceedsTilde]  \[NotPrecedesEqual] ⋫ \[NotRightTriangle] ⋪ \[NotLeftTriangle] ⋭ \[NotRightTriangleEqual] ⋬ \[NotLeftTriangleEqual]  \[NotRightTriangleBar]  \[NotLeftTriangleBar]  \[NotSquareSuperset]  \[NotSquareSubset] ⋣ \[NotSquareSupersetEqual] ⋢ \[NotSquareSubsetEqual]

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.

 form full name alias → \[RightArrow] Esc␣->Esc ← \[LeftArrow] Esc<-Esc ↔ \[LeftRightArrow] Esc<->Esc ⟶ \[LongRightArrow] Esc-->Esc ⟵ \[LongLeftArrow] Esc<--Esc ⟷ \[LongLeftRightArrow] Esc<-->Esc  \[ShortRightArrow]  \[ShortLeftArrow] ↦ \[RightTeeArrow] ↤ \[LeftTeeArrow] ⇥ \[RightArrowBar] ⇤ \[LeftArrowBar] ⇒ \[DoubleRightArrow] Esc␣=>Esc ⇐ \[DoubleLeftArrow] Esc␣<=Esc ⇔ \[DoubleLeftRightArrow] Esc<=>Esc ⟹ \[DoubleLongRightArrow] Esc==>Esc ⟸ \[DoubleLongLeftArrow] Esc<==Esc ⟺ \[DoubleLongLeftRightArrow] Esc<==>Esc
 form full name ↑ \[UpArrow] ↓ \[DownArrow] ↕ \[UpDownArrow] ↥ \[UpTeeArrow] ↧ \[DownTeeArrow] ⤒ \[UpArrowBar] ⤓ \[DownArrowBar] ⇑ \[DoubleUpArrow] ⇓ \[DoubleDownArrow] ⇕ \[DoubleUpDownArrow] ⇄ \[RightArrowLeftArrow] ⇆ \[LeftArrowRightArrow] ⇅ \[UpArrowDownArrow] ⇵ \[DownArrowUpArrow] ↘ \[LowerRightArrow] ↙ \[LowerLeftArrow] ↖ \[UpperLeftArrow] ↗ \[UpperRightArrow]

Ordinary arrows.

 form full name alias ⇀ \[RightVector] EscvecEsc ↼ \[LeftVector] ⥎ \[LeftRightVector] ⇁ \[DownRightVector] ↽ \[DownLeftVector] ⥐ \[DownLeftRightVector] ⥛ \[RightTeeVector] ⥚ \[LeftTeeVector] ⥟ \[DownRightTeeVector] ⥞ \[DownLeftTeeVector] ⥓ \[RightVectorBar] ⥒ \[LeftVectorBar] ⥗ \[DownRightVectorBar] ⥖ \[DownLeftVectorBar] ⇌ \[Equilibrium] EscequiEsc ⇋ \[ReverseEquilibrium]
 form full name ↿ \[LeftUpVector] ⇃ \[LeftDownVector] ⥑ \[LeftUpDownVector] ↾ \[RightUpVector] ⇂ \[RightDownVector] ⥏ \[RightUpDownVector] ⥠ \[LeftUpTeeVector] ⥡ \[LeftDownTeeVector] ⥜ \[RightUpTeeVector] ⥝ \[RightDownTeeVector] ⥘ \[LeftUpVectorBar] ⥙ \[LeftDownVectorBar] ⥔ \[RightUpVectorBar] ⥕ \[RightDownVectorBar] ⥮ \[UpEquilibrium] ⥯ \[ReverseUpEquilibrium]

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.