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Operator Input Forms—Wolfram Language Documentation
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TECH NOTE Characters that are not letters, letter‐like forms, or structural elements are treated by the Wolfram Language as operators. The Wolfram Language 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 backslash—can only be used inside \(…\). The table below gives the interpretations of these operators within ∖!∖(…∖). "Input of Boxes" gives interpretations when no \! is included. expr and expri any expression symb any symbol patt any pattern object string and stringi "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 Objects used in the tables of operator input forms. Operator Precedence operator form full form grouping forms representing numbers (see Numbers ) ⊲ forms representing symbols (see Symbol Names and Contexts ) ⊲ forms representing character strings (see Character Strings ) ⊲ e11 e12 … e21 e22 … … {{e11,e12,…},{e21,e22,…},…} ⊲ e11 e12 e21 e22 … Piecewise[{{e11,e12},{e21,e22},…}] ⊲ expr::string MessageName[expr,"string"] ⊲ expr::string1::string2 MessageName[expr,"string1","string2"] ⊲ forms containing # (see additional input forms) ⊲ forms containing % (see additional input forms) ⊲ forms containing _ (see additional input forms) ⊲ <<filename Get["filename"] ⊲ Overscript[expr1,expr2] expr1\&expr2 Overscript[expr1,expr2] e\&(e\&e) Underscript[expr1,expr2] expr1\+expr2 Underscript[expr1,expr2] e\+(e\+e) Underoverscript[expr1,expr2,expr3] expr1\+expr2\%expr3 Underoverscript[expr1,expr2,expr3] expr1\&expr2\%expr3 Underoverscript[expr1,expr3,expr2] expr1expr2 Subscript[expr1,expr2] e(ee) expr1\_expr2 Subscript[expr1,expr2] e\_(e\_e) expr1\_expr2\%expr3 Power[Subscript[expr1,expr2],expr3] ⊲ \!boxes (interpreted version of boxes ) expr1?expr2 PatternTest[expr1,expr2] ⊲ expr1[expr2,…] expr1[expr2,…] (e[e])[e] ⊲ expr1[[expr2,…]] Part[expr1,expr2,…] (e[[e]])[[e]] ⊲ expr1〚expr2,…〛 Part[expr1,expr2,…] (e〚e〛)〚e〛 ⊲ expr1〚expr2〛 Part[expr1,expr2,…] (e〚e〛)〚e〛 ⊲ expr1::[expr2,…] TypeSpecifier[expr1][expr2,…] (e::[e])::[e] ⊲ \*expr (boxes constructed from expr ) expr++ Increment[expr] ⊲ expr-- Decrement[expr] ⊲ ++expr PreIncrement[expr] ⊲ --expr PreDecrement[expr] ⊲ expr1@*expr2 Composition[expr1,expr2] e@*e@*e ⊲ expr1/*expr2 RightComposition[expr1,expr2] e/*e/*e ⊲ expr1expr2 Application[expr1,expr2] (ee)e expr1@expr2 expr1[expr2] e@(e@e) ⊲ expr1 expr2 (invisible application, input as expr1 Esc@Esc expr2) ⊲ expr1[expr2] expr1~expr2~expr3 expr2[expr1,expr3] (e~e~e)~e~e ⊲ expr1/@expr2 Map[expr1,expr2] e/@(e/@e) ⊲ expr1//@expr2 MapAll[expr1,expr2] e//@(e//@e) ⊲ expr1@@expr2 Apply[expr1,expr2] e@@(e@@e) ⊲ expr1@@@expr2 MapApply[expr1,expr2] e@@@(e@@@e) ⊲ expr! Factorial[expr] ⊲ expr!! Factorial2[expr] ⊲ expr Conjugate[expr] ⊲ expr Transpose[expr] ⊲ expr ConjugateTranspose[expr] ⊲ expr ConjugateTranspose[expr] ⊲ expr' Derivative[1][expr] ⊲ expr''…' (n times) Derivative[n][expr] ⊲ expr1<>expr2<>expr3 StringJoin[expr1,expr2,expr3] e<>e<>e ⊲ expr1^expr2 Power[expr1,expr2] e^(e^e) ⊲ expr1expr2 Power[expr1,expr2] e(ee) ⊲ Power[Subscript[expr1,expr2],expr3] ⊲ expr1\^expr2\%expr3 Power[Subscript[expr1,expr3],expr2] ⊲ vertical arrow and vector operators Sqrt[expr] ⊲ \@ expr Sqrt[expr] \@(\@ e) ⊲ \@ expr\%n Power[expr,1/n] ⊲ expr DifferentialD[expr] ( e) ⊲ ∂expr1expr2 D[expr2,expr1] ∂e(∂ee) ⊲ ∇ expr Del[expr] ∇(∇e) expr1expr2 DiscreteShift[expr2,expr1] e(ee) ⊲ expr1expr2 DiscreteRatio[expr2,expr1] e(ee) ⊲ expr1expr2 DifferenceDelta[expr2,expr1] e(ee) ⊲ expr Square[expr] ( e) expr1∘ expr2∘ expr3 SmallCircle[expr1,expr2,expr3] e∘ e∘ e expr1⊙ expr2⊙ expr3 CircleDot[expr1,expr2,expr3] e ⊙ e ⊙ e expr1**expr2**expr3 NonCommutativeMultiply[expr1,expr2,expr3] e**e**e expr1expr2expr3 Cross[expr1,expr2,expr3] eee ⊲ expr1.expr2.expr3 Dot[expr1,expr2,expr3] e.e.e ⊲ -expr Times[-1,expr] ⊲ +expr expr ⊲ ±expr PlusMinus[expr] ∓expr MinusPlus[expr] expr1/expr2 expr1(expr2)^-1 (e/e)/e ⊲ expr1÷expr2 Divide[expr1,expr2] (e÷e)÷e ⊲ expr1\/expr2 Divide[expr1,expr2] (e\/e)\/e ⊲ expr1∖expr2∖expr3 Backslash[expr1,expr2,expr3] e∖e∖e expr1⋄expr2⋄expr3 Diamond[expr1,expr2,expr3] e⋄e⋄e expr1⋀expr2⋀expr3 Wedge[expr1,expr2,expr3] e⋀e⋀e expr1⋁expr2⋁expr3 Vee[expr1,expr2,expr3] e⋁e⋁e expr1⊗expr2⊗expr3 CircleTimes[expr1,expr2,expr3] e⊗e⊗e expr1·expr2·expr3 CenterDot[expr1,expr2,expr3] e·e·e expr1 expr2 expr3 Times[expr1,expr2,expr3] e e e ⊲ expr1*expr2*expr3 Times[expr1,expr2,expr3] e*e*e ⊲ expr1×expr2×expr3 Times[expr1,expr2,expr3] e×e×e ⊲ expr1⋆expr2⋆expr3 Star[expr1,expr2,expr3] e⋆e⋆e e4 Product[e4,{e1,e2,e3}] ∏(∏ e) ⊲ expr1≀expr2≀expr3 VerticalTilde[expr1,expr2,expr3] e≀e≀e expr1∐expr2∐expr3 Coproduct[expr1,expr2,expr3] e∐e∐e expr1⌢expr2⌢expr3 Cap[expr1,expr2,expr3] e⌢e⌢e expr1⌣expr2⌣expr3 Cup[expr1,expr2,expr3] e⌣e⌣e expr1⊕ expr2⊕ expr3 CirclePlus[expr1,expr2,expr3] e⊕e⊕e expr1⊖ expr2 CircleMinus[expr1,expr2] (e ⊖ e)⊖ e ∫ expr1expr2 Integrate[expr1,expr2] ∫ (∫ e e) e ⊲ e3e4 Integrate[e3,{e4,e1,e2}] ∫ (∫ e e) e ⊲ ∫e1∈e2e3 Integrate[e3,e1∈e2] ∫ (∫ e) ⊲ other integration operators e4 Sum[e4,{e1,e2,e3}] ∑(∑ e) ⊲ e3 Limit[e3,e1e2] (e) ⊲ e3 MaxLimit[e3,e1e2] (e) ⊲ e3 MinLimit[e3,e1e2] (e) ⊲ expr1+expr2+expr3 Plus[expr1,expr2,expr3] e+e+e ⊲ expr1-expr2 expr1+(-1expr2) e-e-e ⊲ expr1±expr2 PlusMinus[expr1,expr2] (e±e)±e expr1∓expr2 MinusPlus[expr1,expr2] (e∓e)∓e expr1⋂expr2 Intersection[expr1,expr2] e⋂e⋂e ⊲ other intersection operators expr1⋃expr2 Union[expr1,expr2] e⋃e⋃e ⊲ other union operators i;;j;;k Span[i,j,k] e;;e;;e ⊲ expr1==expr2 Equal[expr1,expr2] e==e==e ⊲ expr1==expr2 Equal[expr1,expr2] e==e==e ⊲ expr1expr2 Equal[expr1,expr2] eee ⊲ expr1!= expr2 Unequal[expr1,expr2] e!=e!=e ⊲ expr1!=expr2 Unequal[expr1,expr2] e!=e!=e ⊲ other equality and similarity operators expr1>expr2 Greater[expr1,expr2] e>e>e ⊲ expr1>=expr2 GreaterEqual[expr1,expr2] e>=e>=e ⊲ expr1≥expr2 GreaterEqual[expr1,expr2] e≥e≥e ⊲ expr1⩾expr2 GreaterEqual[expr1,expr2] e⩾e⩾e ⊲ expr1<expr2 Less[expr1,expr2] e<e<e ⊲ expr1<=expr2 LessEqual[expr1,expr2] e<=e<=e ⊲ expr1≤expr2 LessEqual[expr1,expr2] e≤e≤e ⊲ expr1⩽expr2 LessEqual[expr1,expr2] e⩽e⩽e ⊲ other ordering operators expr1expr2 VerticalBar[expr1,expr2] eee expr1expr2 NotVerticalBar[expr1,expr2] eee expr1∥expr2 DoubleVerticalBar[expr1,expr2] e∥e∥e expr1∦expr2 NotDoubleVerticalBar[expr1,expr2] e∦e∦e horizontal arrow and vector operators diagonal arrow operators expr1===expr2 SameQ[expr1,expr2] e===e===e ⊲ expr1=!=expr2 UnsameQ[expr1,expr2] e=!=e=!=e ⊲ expr1∈expr2 Element[expr1,expr2] e∈e∈e ⊲ expr1∉expr2 NotElement[expr1,expr2] e∉e∉e ⊲ expr1⊂expr2 Subset[expr1,expr2] e⊂e⊂e expr1⊃expr2 Superset[expr1,expr2] e⊃e⊃e other set relation operators ∀expr1expr2 ForAll[expr1,expr2] ∀e(∀ee) ⊲ ∃expr1expr2 Exists[expr1,expr2] ∃e(∃ee) ⊲ ∄expr1expr2 NotExists[expr1,expr2] ∄e(∄ee) !expr Not[expr] !(!e) ⊲ ¬expr Not[expr] ¬(¬e) ⊲ expr1&&expr2&&expr3 And[expr1,expr2,expr3] e&&e&&e ⊲ expr1∧expr2∧expr3 And[expr1,expr2,expr3] e∧e∧e ⊲ expr1⊼expr2⊼expr3 Nand[expr1,expr2,expr3] e⊼e⊼e ⊲ expr1⊻expr2⊻expr3 Xor[expr1,expr2,expr3] e⊻e⊻e ⊲ expr1expr2expr3 Xnor[expr1,expr2,expr3] eee ⊲ expr1||expr2||expr3 Or[expr1,expr2,expr3] e||e||e ⊲ expr1∨expr2∨expr3 Or[expr1,expr2,expr3] e∨e∨e ⊲ expr1⊽expr2⊽expr3 Nor[expr1,expr2,expr3] e⊽e⊽e ⊲ expr1⧦expr2⧦expr3 Equivalent[expr1,expr2,expr3] e⧦e⧦e ⊲ expr1expr2 Implies[expr1,expr2] e(ee) ⊲ expr1⥰expr2 Implies[expr1,expr2] e⥰e⥰e ⊲ expr1⊢expr2 RightTee[expr1,expr2] e⊢(e⊢e) expr1⊨expr2 DoubleRightTee[expr1,expr2] e⊨(e⊨e) expr1⊣expr2 LeftTee[expr1,expr2] (e⊣e)⊣e expr1⫤expr2 DoubleLeftTee[expr1,expr2] (e⫤e)⫤e expr1⊥expr2 UpTee[expr1,expr2] (e⊥e)⊥e expr1⊤expr2 DownTee[expr1,expr2] (e⊤e)⊤e expr1∍expr2 SuchThat[expr1,expr2] e∍(e∍e) expr.. Repeated[expr] ⊲ expr... RepeatedNull[expr] ⊲ expr1|expr2 Alternatives[expr1,expr2] e|e|e ⊲ symb:expr Pattern[symb,expr] ⊲ symb:patt:expr Optional[Pattern[symb,patt],expr] ⊲ patt:expr Optional[patt,expr] ⊲ expr1~~expr2~~expr3 StringExpression[expr1,expr2,expr3] e~~e~~e ⊲ expr1/;expr2 Condition[expr1,expr2] (e/;e)/;e ⊲ expr1<->expr2 TwoWayRule[expr1,expr2] e<->(e<->e) ⊲ expr1expr2 TwoWayRule[expr1,expr2] e(ee) ⊲ expr1expr2 Rule[expr1,expr2] e(ee) ⊲ expr1expr2 Rule[expr1,expr2] e(ee) ⊲ expr1:>expr2 RuleDelayed[expr1,expr2] e:>(e:>e) ⊲ expr1 expr2 RuleDelayed[expr1,expr2] e(ee) ⊲ expr1/.expr2 ReplaceAll[expr1,expr2] (e/.e)/.e ⊲ expr1//.expr2 ReplaceRepeated[expr1,expr2] (e//.e)//.e ⊲ expr1+=expr2 AddTo[expr1,expr2] e+=(e+=e) ⊲ expr1-=expr2 SubtractFrom[expr1,expr2] e-=(e-=e) ⊲ expr1*=expr2 TimesBy[expr1,expr2] e*=(e*=e) ⊲ expr1/=expr2 DivideBy[expr1,expr2] e/=(e/=e) ⊲ expr& Function[expr] ⊲ expr1∶expr2 Colon[expr1,expr2] e∶e∶e expr1//=expr2 ApplyTo[expr1,expr2] e//=(e//=e) ⊲ expr1//expr2 expr2[expr1] (e//e)//e expr1expr2 VerticalSeparator[expr1,expr2] eee expr1∴expr2 Therefore[expr1,expr2] e∴(e∴e) expr1∵expr2 Because[expr1,expr2] (e∵e)∵e expr1=expr2 Set[expr1,expr2] e=(e=e) ⊲ expr1:=expr2 SetDelayed[expr1,expr2] e:=(e:=e) ⊲ expr1^=expr2 UpSet[expr1,expr2] e^=(e^=e) ⊲ expr1^:=expr2 UpSetDelayed[expr1,expr2] e^:=(e^:=e) ⊲ symb/:expr1=expr2 TagSet[symb,expr1,expr2] ⊲ symb/:expr1:=expr2 TagSetDelayed[symb,expr1,expr2] ⊲ expr=. Unset[expr] ⊲ symb/:expr=. TagUnset[symb,expr] ⊲ expr1|->expr2 Function[expr1,expr2] e(ee) ⊲ expr1expr2 Function[expr1,expr2] e(ee) ⊲ expr>>filename Put[expr,"filename"] ⊲ expr>>>filename PutAppend[expr,"filename"] ⊲ expr1;expr2;expr3 CompoundExpression[expr1,expr2,expr3] ⊲ expr1;expr2; CompoundExpression[expr1,expr2,Null] ⊲ expr1\`expr2 FormBox[expr2,expr1] e\`(e\`e) ⊲ Operator input forms, in order of decreasing precedence. Operators of equal precedence are grouped together. special input form full form # Slot[1] #n Slot[n] #string Slot["string"] ## SlotSequence[1] ##n SlotSequence[n] % Out[ ] %% Out[-2] %%…% ( n times) Out[-n] %n Out[n] _ Blank[ ] _expr Blank[expr] __ BlankSequence[ ] __expr BlankSequence[expr] ___ BlankNullSequence[ ] ___expr BlankNullSequence[expr] _. Optional[Blank[ ]] symb_ Pattern[symb,Blank[ ]] symb_expr Pattern[symb,Blank[expr]] symb__ Pattern[symb,BlankSequence[ ]] symb__expr Pattern[symb,BlankSequence[expr]] symb___ Pattern[symb,BlankNullSequence[ ]] symb___expr Pattern[symb,BlankNullSequence[expr]] symb_. Optional[Pattern[symb,Blank[ ]]] _ :expr Optional[Blank[],expr] symb_:expr Optional[Pattern[symb,Blank[]],expr] _head :expr Optional[Blank[head],expr] symb_head:expr Optional[Pattern[symb,Blank[head]],expr] Additional input forms, in order of decreasing precedence. Special Characters Special characters that appear in operators usually have names that correspond to the names of the functions they represent. Thus the character has the name ⊕ and yields the function CirclePlus. Exceptions are ⩾, ⩽ and ⥰. The delimiters in matchfix operators have names \[LeftName] and \[RightName]. "Listing of Named Characters" gives a complete listing of special characters that appear in operators. keyboard characters special character -> :> == != ≠ ≠ keyboard characters special character >= ≥ ≥ >= ⩾ ⩾ <= ≤ ≤ <= ⩽ ⩽ Keyboard and special characters with the same interpretations. keyboard character special character : : ∶ ∶ ~ ~ ∼ ∼ ^ ^ ⋀ ⋀ ^ ^ ∧ ∧ * * ⋆ ⋆ \ ∖ ∖ ∖ keyboard character special character . . · · | | | | | | - - – – ... … … Some keyboard and special characters with different interpretations. Precedence and the Ordering of Input Forms 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 ∮, ∲ and ∯ work the same as ∫. See "Two-Dimensional Input Forms" for two‐dimensional input forms associated with integration operators. Spaces and Multiplication Spaces in the Wolfram Language denote multiplication, just as they do in standard mathematical notation. In addition, the Wolfram Language 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 Alternative forms for multiplication. 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 Some cases to be careful about. 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_, the Wolfram Language 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 ( ) ■ , , …, ■ , , …, ■ ␣ (␣) Spacing characters equivalent to an ordinary keyboard space. 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 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 ), ], or }, as well as semicolons and commas.
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