DomainAndRange[Function[{Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, d]}, Signature[{Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, d]}]]] == PowerSet[Integers, n] \[LongRightArrow] {-1, 0, 1}

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8902; </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> * </mo> <mo> &#8230; </mo> <mo> * </mo> <msub> <mi> n </mi> <mi> d </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10230; </mo> <msub> <mi> &#949; </mi> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mtext> </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mtext> </mtext> <mo> , </mo> <mtext> </mtext> <msub> <mi> n </mi> <mi> d </mi> </msub> </mrow> </msub> </mrow> <mo> &#8759; </mo> <mrow> <msup> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Reals]], Function[Rationals]], Function[Reals]] </annotation> </semantics> <mi> n </mi> </msup> <mo> &#10230; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportion </ci> <apply> <ci> LongRightArrow </ci> <apply> <ci> Star </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> d </ci> </apply> </apply> </apply> <apply> <ci> Signature </ci> <list> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> d </ci> </apply> </list> </apply> </apply> <apply> <ci> LongRightArrow </ci> <apply> <power /> <reals /> <ci> n </ci> </apply> <list> <cn type='integer'> -1 </cn> <cn type='integer'> 0 </cn> <cn type='integer'> 1 </cn> </list> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DomainAndRange", "[", RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "d_"]]], "}"]], ",", RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "d_"]]], "}"]], "]"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["PowerSet", "[", RowBox[List["Integers", ",", "n"]], "]"]], "\[LongRightArrow]", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0", ",", "1"]], "}"]]]]]]]]

2001-10-29