## 3.10.4 Operators

### Basic Mathematical Operators

 form full name alias × \[Times] * ÷ \[Divide] div \[Sqrt] sqrt
 form full name alias \[Cross] cross ± \[PlusMinus] +- \[MinusPlus] -+

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] del \[PartialD] pd \[DifferentialD] dd \[Sum] sum \[Product] prod
 form full name alias \[Integral] int \[ContourIntegral] cint \[DoubleContourIntegral] \[CounterClockwiseContourIntegral] cccint \[ClockwiseContourIntegral] ccint

Operators used in calculus.

 form full name aliases  \[Conjugate] co, conj  \[Transpose] tr
 form full name alias  \[ConjugateTranspose] ct  \[HermitianConjugate] hc

Operators for complex numbers and matrices.

### 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] :

Operators used as logical connectives.

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].

This is interpreted using the built-in functions And and Implies.
 In[1]:=  3 < 4 x > 5 y < 7
 Out[1]=

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

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

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.

The operators are interpreted as functions with corresponding names.
 In[1]:=  x y z // FullForm
 Out[1]//FullForm=

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||

Characters used as bracketing operators.

 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, ... ]

Interpretations of bracketing operators.

### 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]

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

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

Operators usually used for ordering by magnitude.

 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

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

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 ->.
 In[1]:=  x + y /. x -> 3
 Out[1]=

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

Arrow-like operators with built-in meanings in Mathematica.

 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]

Ordinary arrows.

 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]

Vectors and related arrows.
All the arrow and vector-like operators in Mathematica are infix.
 In[2]:=  x y z
 Out[2]=

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

Tees.

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