A.2.7 Operator Input Forms

Characters that are not letters, letter-like forms or structural elements are treated by Mathematica as operators. Mathematica has built-in rules for interpreting all operators. The functions to which these operators correspond may or may not, however, have built-in evaluation or other rules. Cases in which built-in meanings are by default defined are indicated by in the tables below.

Operators that construct two-dimensional boxes--all of which have names beginning with back-slash--can only be used inside \( ... \). The table below gives the interpretations of these operators within \!\( ... \). Section A.2.9 gives interpretations when no \! is included.

expr and	any expression
symb	any symbol
patt	any pattern object
string and	"cccc" or a sequence of letters, letter-like forms and digits
filename	like string, but can include additional characters described below
	built-in meanings exist

operator form	full form	grouping
forms representing numbers (see Section A.2.5)		
forms representing symbols (see Section A.2.4)		
forms representing character strings (see Section A.2.3)		
	{{, , ... }, {, , ... }, ... }
	Piecewise[{{, }, {, }, ... }]
expr::string	MessageName[expr, "string"]
expr::::	MessageName[expr, "", ""]
forms containing # (see Section A.2.7)		
forms containing % (see Section A.2.7)		
forms containing _ (see Section A.2.7)		
<< filename	Get["filename"]
	Overscript[, ]
\&	Overscript[, ]	e\&(e\&e)
	Underscript[, ]
\+	Underscript[, ]	e\+(e\+e)
	Underoverscript[, , ]
\+ \%	Underoverscript[, , ]
\& \%	Underoverscript[, , ]
	Subscript[, ]
\_	Subscript[, ]	e\_(e\_e)
\_ \%	Power[Subscript[, ], ]
\!boxes	(interpreted version of boxes)
?	PatternTest[, ]
[, ... ]	[, ... ]	(e[e])[e]
[[, ... ]]	Part[, , ... ]	(e[[e]])[[e]]
, ...	Part[, , ... ]	(ee)e
	Part[, , ... ]

operator form	full form	grouping
\*expr	(boxes constructed from expr)
expr++	Increment[expr]
expr--	Decrement[expr]
++expr	PreIncrement[expr]
--expr	PreDecrement[expr]
@	[]	e @ (e @ e)
	(invisible application, input as @ )
	[]
~ ~	[, ]	(e ~ e ~ e) ~ e ~ e
/@	Map[, ]	e /@ (e /@ e)
//@	MapAll[, ]	e //@ (e //@ e)
@@	Apply[, ]	e @@ (e @@ e)
@@@	Apply[, , {1}]	e @@@ (e @@@ e)
expr!	Factorial[expr]
expr!!	Factorial2[expr]
	Conjugate[expr]
	Transpose[expr]
	ConjugateTranspose[expr]
	ConjugateTranspose[expr]
expr'	Derivative[1][expr]
expr'' ... ' (n times)	Derivative[n][expr]
<> <>	StringJoin[, , ]	e <> e <> e
^	Power[, ]	e^(e^e)
	Power[, ]
	Power[Subscript[, ], ]
\^ \%	Power[Subscript[, ], ]
vertical arrow and vector operators		
expr	Sqrt[expr]	(e)
\@ expr	Sqrt[expr]	\@(\@e)
\@ expr \% n	Power[expr, 1/n]
	Integrate[, ]	( e e) e
	Integrate[, {, , }]	( e e) e
other integration operators: see Section A.2.7		
	D[, ]
expr	Del[expr]	( e)
expr	Square[expr]	( e)
	SmallCircle[, , ]	e e e

operator form	full form	grouping
	CircleDot[, , ]	e e e
** **	NonCommutativeMultiply[, , ]	e ** e ** e
	Cross[, , ]	e e e
. .	Dot[, , ]	e . e . e
-expr	Times[-1, expr]
+expr	expr
expr	PlusMinus[expr]
expr	MinusPlus[expr]
/	()^-1	(e / e) / e
	Divide[, ]	(e e) e
\/	Divide[, ]	(e\/e)\/e
	Backslash[, , ]	e e e
	Diamond[, , ]	e e e
	Wedge[, , ]	e e e
	Vee[, , ]	e e e
	CircleTimes[, , ]	e e e
	CenterDot[, , ]	e e e
	Times[, , ]	e e e
* *	Times[, , ]	e * e * e
	Times[, , ]	e e e
	Star[, , ]	e e e
	Product[, {, , }]	( e)
	VerticalTilde[, , ]	e e e
	Coproduct[, , ]	e e e
	Cap[, , ]	e e e
	Cup[, , ]	e e e
	CirclePlus[, , ]	e e e
	CircleMinus[, ]	(e e) e

operator form	full form	grouping
	Sum[, {, , }]	( e)
+ +	Plus[, , ]	e + e + e
-	+ (-1 )	(e - e) - e
	PlusMinus[, ]	(e e) e
	MinusPlus[, ]	(e e) e
	Intersection[, ]	e e e
other intersection operators		
	Union[, ]	e e e
other union operators		
	Equal[, ]	e e e
	Equal[, ]	e e e
	Equal[, ]	e e e
	Unequal[, ]	e e e
	Unequal[, ]	e e e
other equality and similarity operators		
>	Greater[, ]	e > e > e
	GreaterEqual[, ]	e e e
	GreaterEqual[, ]	e e e
	GreaterEqual[, ]	e e e
<	Less[, ]	e < e < e
	LessEqual[, ]	e e e
	LessEqual[, ]	e e e
	LessEqual[, ]	e e e
other ordering operators		
	VerticalBar[, ]	e e e
	NotVerticalBar[, ]	e e e
	DoubleVerticalBar[, ]	e e e
	NotDoubleVerticalBar[, ]	e e e
horizontal arrow and vector operators		
diagonal arrow operators		
=	SameQ[, ]	e = e = e
=	UnsameQ[, ]	e = e = e

operator form	full form	grouping
	Element[, ]	e e e
	NotElement[, ]	e e e
	Subset[, ]	e e e
	Superset[, ]	e e e
other set relation operators		
	ForAll[, ]
	Exists[, ]
	NotExists[, ]
!expr	Not[expr]	!(!e)
expr	Not[expr]	( e)
&& &&	And[, , ]	e && e && e
	And[, , ]	e e e
	Nand[, , ]	e e e
	Xor[, , ]	e e e
|| ||	Or[, , ]	e || e || e
	Or[, , ]	e e e
	Nor[, , ]	e e e
	Implies[, ]	e(ee)
	Implies[, ]	e(ee)
	RightTee[, ]	e (e e)
	DoubleRightTee[, ]	e (e e)
	LeftTee[, ]	(e e) e
	DoubleLeftTee[, ]	(e e) e
	SuchThat[, ]	e (e e)
expr..	Repeated[expr]
expr...	RepeatedNull[expr]
|	Alternatives[, ]	e | e | e
symb:expr	Pattern[symb, expr]
patt:expr	Optional[patt, expr]

operator form	full form	grouping
~~ ~~	StringExpression[, , ]	e ~~ e ~~ e
/;	Condition[, ]	(e/;e)/;e
->	Rule[, ]	e -> (e -> e)
	Rule[, ]	e (e e)
:>	RuleDelayed[, ]	e :> (e :> e)
	RuleDelayed[, ]	e (e e)
/.	ReplaceAll[, ]	(e /. e) /. e
//.	ReplaceRepeated[, ]	(e //. e) //. e
+=	AddTo[, ]	e += (e += e)
-=	SubtractFrom[, ]	e -= (e -= e)
*=	TimesBy[, ]	e *= (e *= e)
/=	DivideBy[, ]	e /= (e /= e)
expr &	Function[expr]
	Colon[, ]	e e e
//	[]	(e // e) // e
	VerticalSeparator[, ]	e e e
	Therefore[, ]	e (e e)
	Because[, ]	(e e) e
=	Set[, ]	e = (e = e)
:=	SetDelayed[, ]	e := (e := e)
^=	UpSet[, ]	e ^= (e ^= e)
^:=	UpSetDelayed[, ]	e ^:= (e ^:= e)
symb/: =	TagSet[symb, , ]
symb/: :=	TagSetDelayed[symb, , ]
expr =.	Unset[expr]
symb/: expr =.	TagUnset[symb, expr]
expr >> filename	Put[expr, "filename"]
expr >>> filename	PutAppend[expr, "filename"]
;;	CompoundExpression[, , ]
;;	CompoundExpression[, , Null]
\`	FormBox[, ]	e \` (e \` e)

Section A.12.1 gives a complete listing of special characters that appear in operators.

keyboard characters special character -> \[Rule] :> \[RuleDelayed] \[Equal] \[NotEqual]

keyboard characters special character \[GreaterEqual] \[GreaterSlantEqual] \[LessEqual] \[LessSlantEqual]

keyboard character special character \[RawColon] : \[Colon] \[RawTilde] ~ \[Tilde] \[RawWedge] ^ \[Wedge] \[RawWedge] ^ \[And] \[RawStar] * \[Star] \[RawBackslash] \ \[Backslash]

keyboard character special character \[RawDot] . \[CenterDot] \[RawVerticalBar] | \[VerticalBar] \[RawVerticalBar] | \[VerticalSeparator] \[RawVerticalBar] | \[LeftBracketingBar] \[RawDash] - \[Dash] ... \[Ellipsis]

Precedence and the Ordering of Input Forms

The tables of input forms are arranged in decreasing order of precedence. Input forms in the same box have the same precedence. Each page in the table begins a new box. As discussed in Section 2.1.3, precedence determines how Mathematica groups terms in input expressions. The general rule is that if has higher precedence than , then is interpreted as , and is interpreted as .

Grouping of Input Forms

The third columns in the tables show how multiple occurrences of a single input form, or of several input forms with the same precedence, are grouped. For example, a/b/c is grouped as (a/b)/c ("left associative"), while a^b^c is grouped as a^(b^c) ("right associative"). No grouping is needed in an expression like a + b + c, since Plus is fully associative, as represented by the attribute Flat.

Precedence of Integration Operators

Forms such as have an "outer" precedence just below Power, as indicated in the table above, but an "inner" precedence just above . The outer precedence determines when needs to be parenthesized; the inner precedence determines when needs to be parenthesized.

\[ContourIntegral], \[ClockwiseContourIntegral] and \[DoubleContourIntegral] work the same as \[Integral].

See Section A.2.8 for two-dimensional input forms associated with integration operators.

Spaces and Multiplication

Spaces in Mathematica denote multiplication, just as they do in standard mathematical notation. In addition, Mathematica takes complete expressions that are adjacent, not necessarily separated by spaces, to be multiplied together.

• x y z x*y*z

• 2x 2*x

• 2(x+1) 2*(x+1)

• c(x+1) c*(x+1)

• (x+1)(y+2) (x+1)*(y+2)

• x! y x!*y

• x!y x!*y

An expression like x!y could potentially mean either (x!)*y or x*(!y). The first interpretation is chosen because Factorial has higher precedence than Not.

Spaces within single input forms are ignored. Thus, for example, a + b is equivalent to a+b. You will often want to insert spaces around lower precedence operators to improve readability.

You can give a "coefficient" for a symbol by preceding it with any sequence of digits. When you use numbers in bases larger than 10, the digits can include letters. (In bases other than 10, there must be a space between the end of the coefficient, and the beginning of the symbol name.)

• x^2y, like x^2 y, means (x^2) y

• x/2y, like x/2 y, means (x/2) y

• xy is a single symbol, not x*y

Spaces to Avoid

You should avoid inserting any spaces between the different characters in composite operators such as /., =. and . Although in some cases such spaces are allowed, they are liable to lead to confusion.

Another case where spaces must be avoided is between the characters of the pattern object x_. If you type x _, Mathematica will interpret this as x*_, rather than the single named pattern object x_.

Similarly, you should not insert any spaces inside pattern objects like x_:value.

Spacing Characters

• Ordinary keyboard space (\[RawSpace])

• \[VeryThinSpace], \[ThinSpace], ..., \[ThickSpace]

• \[NegativeVeryThinSpace], \[NegativeThinSpace], ..., \[NegativeThickSpace]

• (\[SpaceIndicator])

Relational Operators

Relational operators can be mixed. An expression like a > b c is converted to Inequality[a, Greater, b, GreaterEqual, c], which effectively evaluates as (a > b) && (b c). (The reason for the intermediate Inequality form is that it prevents objects from being evaluated twice when input like a > b c is processed.)

File Names

Any file name can be given in quotes after <<, >> and >>>. File names can also be given without quotes if they contain only alphanumeric characters, special characters and the characters `, /, ., \, !, -, _, :, $, *, ~ and ?, together with matched pairs of square brackets enclosing any characters other than spaces, tabs and newlines. Note that file names given without quotes can be followed only by spaces, tabs or newlines, or by the characters ), ], as well as semicolon and comma.