3.10.4 Operators

Basic Mathematical Operators

form full name alias × \[Times] * ÷ \[Divide] div \[Sqrt] sqrt

form full name alias \[Cross] cross ± \[PlusMinus] +- \[MinusPlus] -+

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)

Operators in Calculus and Algebra

form full name alias \[Del] del \[PartialD] pd \[DifferentialD] dd \[Sum] sum \[Product] prod

form full name alias \[Integral] int \[ContourIntegral] cint \[DoubleContourIntegral] \[CounterClockwiseContourIntegral] cccint \[ClockwiseContourIntegral] ccint

form full name aliases  \[Conjugate] co, conj  \[Transpose] tr

form full name alias  \[ConjugateTranspose] ct  \[HermitianConjugate] hc

Logical and Other Connectives

form full name aliases \[And] &&, and \[Or] ||, or ¬ \[Not] !, not \[Element] el \[ForAll] fa \[Exists] ex \[NotExists] !ex \[Xor] xor \[Nand] nand \[Nor] nor

form full name alias \[Implies] => \[RoundImplies] \[Therefore] tf \[Because] \[RightTee] \[LeftTee] \[DoubleRightTee] \[DoubleLeftTee] \[SuchThat] st \[VerticalSeparator] | : \[Colon] :

The operators , and are interpreted as corresponding to the built-in functions And, Or and Not, and are equivalent to the keyboard operators &&, || and !. The operators , and correspond to the built-in functions Xor, Nand and Nor. Note that is a prefix operator.

xy and xy are both taken to give the built-in function Implies[x, y]. xy gives the built-in function Element[x, y].

Mathematica supports most of the standard syntax used in mathematical logic. In Mathematica, 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 .

In[2]:=  XMLElement[img, {src -> http://documents.wolfram.com/MathematicaCharacters/ForAll.gif, width -> 7, height -> 19, align -> absmiddle, alt -> ForAll}, {}] . XMLElement[sub, {}, {x}]XMLElement[img, {src -> http://documents.wolfram.com/MathematicaCharacters/Exists.gif, width -> 7, height -> 19, align -> absmiddle, alt -> Exists}, {}] . XMLElement[sub, {}, {y}][x,y]//FullForm Out[2]//FullForm=

Operators Used to Represent Actions

form full name alias \[SmallCircle] sc \[CirclePlus] c+ \[CircleMinus] c- \[CircleTimes] c* \[CircleDot] c. \[Diamond] dia \[CenterDot] . \[Star] star \[VerticalTilde] \ \[Backslash] \

form full name alias \[Wedge] ^ \[Vee] v \[Union] un \[UnionPlus] \[Intersection] inter \[SquareIntersection] \[SquareUnion] \[Coproduct] coprod \[Cap] \[Cup] \[Square] sq

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

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] lf \[RightFloor] rf \[LeftCeiling] lc \[RightCeiling] rc \[LeftDoubleBracket] [[ \[RightDoubleBracket] ]]
	form full name alias \[LeftAngleBracket] < \[RightAngleBracket] > \[LeftBracketingBar] l| \[RightBracketingBar] r| \[LeftDoubleBracketingBar] l|| \[RightDoubleBracketingBar] r||

x	Floor[x]
x	Ceiling[x]
mi,j, ...	Part[m, i, j, ... ]
x,y, ...	AngleBracket[x, y, ... ]
x,y, ...	BracketingBar[x, y, ... ]
x,y, ...	DoubleBracketingBar[x, y, ... ]

Operators Used to Represent Relations

form full name alias ⩵ \[Equal] \[LongEqual] l= \[Congruent] = \[Tilde] \[TildeTilde] \[TildeEqual] = \[TildeFullEqual] \[EqualTilde] = \[HumpEqual] h= \[HumpDownHump] \[CupCap] \[DotEqual]

form full name alias ≠ \[NotEqual] \[NotCongruent] \[NotTilde] ! \[NotTildeTilde] ! \[NotTildeEqual] != \[NotTildeFullEqual] ! \[NotEqualTilde] \[NotHumpEqual] !h= \[NotHumpDownHump] \[NotCupCap] \[Proportional] prop \[Proportion]

The special character (or \[Equal]) is an alternative input form for .  is used both for input and output.

In[1]:=  {a b, a ⩵ b, a b, a ≠ b} Out[1]=

form full name alias ≥ \[GreaterEqual] ≤ \[LessEqual] \[GreaterSlantEqual] >/ \[LessSlantEqual] </ \[GreaterFullEqual] \[LessFullEqual] \[GreaterTilde] > \[LessTilde] < \[GreaterGreater] \[LessLess] \[NestedGreaterGreater] \[NestedLessLess] \[GreaterLess] \[LessGreater] \[GreaterEqualLess] \[LessEqualGreater]
	form full name alias \[NotGreaterEqual] ! \[NotLessEqual] ! \[NotGreaterSlantEqual] !>/ \[NotLessSlantEqual] !</ \[NotGreaterFullEqual] \[NotLessFullEqual] \[NotGreaterTilde] !> \[NotLessTilde] !< \[NotGreaterGreater] \[NotLessLess] \[NotNestedGreaterGreater] \[NotNestedLessLess] \[NotGreaterLess] \[NotLessGreater] \[NotGreater] !> \[NotLess] !<

form full name alias \[Subset] sub \[Superset] sup \[SubsetEqual] sub= \[SupersetEqual] sup= \[Element] el \[ReverseElement] mem

form full name alias \[NotSubset] !sub \[NotSuperset] !sup \[NotSubsetEqual] !sub= \[NotSupersetEqual] !sup= \[NotElement] !el \[NotReverseElement] !mem

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]

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

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 ->.

In[1]:=  x + y /. x -> 3 Out[1]=

form full name alias -> \[Rule] -> :> \[RuleDelayed] :>

form full name alias \[Implies] => \[RoundImplies]

form full name alias \[RightArrow] -> \[LeftArrow] <- \[LeftRightArrow] <-> \[LongRightArrow] --> \[LongLeftArrow] <-- \[LongLeftRightArrow] <--> \[ShortRightArrow] \[ShortLeftArrow] \[RightTeeArrow] \[LeftTeeArrow] \[RightArrowBar] \[LeftArrowBar] \[DoubleRightArrow] => \[DoubleLeftArrow] \[DoubleLeftRightArrow] > \[DoubleLongRightArrow] > \[DoubleLongLeftArrow] = \[DoubleLongLeftRightArrow] =>
	form full name \[UpArrow] \[DownArrow] \[UpDownArrow] \[UpTeeArrow] \[DownTeeArrow] \[UpArrowBar] \[DownArrowBar] \[DoubleUpArrow] \[DoubleDownArrow] \[DoubleUpDownArrow] \[RightArrowLeftArrow] \[LeftArrowRightArrow] \[UpArrowDownArrow] \[DownArrowUpArrow] \[LowerRightArrow] \[LowerLeftArrow] \[UpperLeftArrow] \[UpperRightArrow]

form full name alias \[RightVector] vec \[LeftVector] \[LeftRightVector] \[DownRightVector] \[DownLeftVector] \[DownLeftRightVector] \[RightTeeVector] \[LeftTeeVector] \[DownRightTeeVector] \[DownLeftTeeVector] \[RightVectorBar] \[LeftVectorBar] \[DownRightVectorBar] \[DownLeftVectorBar] \[Equilibrium] equi \[ReverseEquilibrium]
	form full name \[LeftUpVector] \[LeftDownVector] \[LeftUpDownVector] \[RightUpVector] \[RightDownVector] \[RightUpDownVector] \[LeftUpTeeVector] \[LeftDownTeeVector] \[RightUpTeeVector] \[RightDownTeeVector] \[LeftUpVectorBar] \[LeftDownVectorBar] \[RightUpVectorBar] \[RightDownVectorBar] \[UpEquilibrium] \[ReverseUpEquilibrium]

form full name alias \[RightTee] rT \[LeftTee] lT \[UpTee] uT \[DownTee] dT

form full name \[DoubleRightTee] \[DoubleLeftTee]