Alternating Function
📂Functions Alternating Function Definition Let set X X X alternating function .
ϕ : X × X × ⋯ × X ⏞ n → R ϕ ( x 1 , … , x i , x i + 1 , … , x n ) = − ϕ ( x 1 , … , x i + 1 , x i , … , x n )
\phi : \overbrace{X \times X \times \cdots \times X}^{n} \to \mathbb{R}
\\ \phi (x_{1}, \dots, x_{i}, x_{i+1}, \dots, x_{n}) = - \phi (x_{1}, \dots, x_{i+1}, x_{i}, \dots, x_{n})
ϕ : X × X × ⋯ × X n → R ϕ ( x 1 , … , x i , x i + 1 , … , x n ) = − ϕ ( x 1 , … , x i + 1 , x i , … , x n )
Explanation It is a function whose sign changes when two adjacent variables are swapped. Of course, it can also be shown that this holds true for variables that are not adjacent, according to the definition.
If there is at least one duplicate element among the variables, the value of the function is 0 0 0
Properties ϕ ( x 1 , … , x i , … , x i + k , … , x n ) = − ϕ ( x 1 , … , x i + k , … , x i , … , x n )
\begin{equation}
\phi (x_{1}, \dots, x_{i}, \dots, x_{i+k}, \dots, x_{n}) = - \phi (x_{1}, \dots, x_{i+k}, \dots, x_{i}, \dots, x_{n})
\end{equation}
ϕ ( x 1 , … , x i , … , x i + k , … , x n ) = − ϕ ( x 1 , … , x i + k , … , x i , … , x n )
The necessary and sufficient condition for ϕ \phi ϕ
ϕ ( x 1 , … , x i , x i , … , x n ) = 0
\begin{equation}
\phi (x_{1}, \dots, x_{i}, x_{i}, \dots, x_{n}) = 0
\end{equation}
ϕ ( x 1 , … , x i , x i , … , x n ) = 0
Proof Proof (1) ϕ ( x 1 , … , x i , … , x i + k − 1 , x i + k , … , x n ) = − ϕ ( x 1 , … , x i , … , x i + k , x i + k − 1 , … , x n ) = ( − 1 ) 2 ϕ ( x 1 , … , x i , … , x i + k , x i + k − 2 , x i + k − 1 , … , x n ) ⋮ = ( − 1 ) k ϕ ( x 1 , … , x i + k , x i , … , x i + k − 2 , x i + k − 1 , … , x n ) = ( − 1 ) k + 1 ϕ ( x 1 , … , x i + k , x i + 1 , x i , … , x i + k − 2 , x i + k − 1 , … , x n ) ⋮ = ( − 1 ) k + ( k − 1 ) ϕ ( x 1 , … , x i + k , … , x i + k − 1 , x i , … , x n ) = − ϕ ( x 1 , … , x i + k , … , x i + k − 1 , x i , … , x n )
\begin{align*}
& \phi (x_{1}, \dots, x_{i}, \dots, x_{i+k-1}, x_{i+k}, \dots, x_{n})
\\ =&\ - \phi (x_{1}, \dots, x_{i}, \dots, x_{i+k}, x_{i+k-1}, \dots, x_{n})
\\ =&\ (-1)^{2} \phi (x_{1}, \dots, x_{i}, \dots, x_{i+k}, x_{i+k-2}, x_{i+k-1}, \dots, x_{n})
\\ \vdots&
\\ =&\ (-1)^{k} \phi (x_{1}, \dots, x_{i+k}, x_{i}, \dots, x_{i+k-2}, x_{i+k-1}, \dots, x_{n})
\\ =&\ (-1)^{k+1} \phi (x_{1}, \dots, x_{i+k}, x_{i+1} ,x_{i}, \dots, x_{i+k-2}, x_{i+k-1}, \dots, x_{n})
\\ \vdots&
\\ =&\ (-1)^{k+(k-1)} \phi (x_{1}, \dots, x_{i+k}, \dots, x_{i+k-1}, x_{i}, \dots, x_{n})
\\ =&\ - \phi (x_{1}, \dots, x_{i+k}, \dots, x_{i+k-1}, x_{i}, \dots, x_{n})
\end{align*}
= = ⋮ = = ⋮ = = ϕ ( x 1 , … , x i , … , x i + k − 1 , x i + k , … , x n ) − ϕ ( x 1 , … , x i , … , x i + k , x i + k − 1 , … , x n ) ( − 1 ) 2 ϕ ( x 1 , … , x i , … , x i + k , x i + k − 2 , x i + k − 1 , … , x n ) ( − 1 ) k ϕ ( x 1 , … , x i + k , x i , … , x i + k − 2 , x i + k − 1 , … , x n ) ( − 1 ) k + 1 ϕ ( x 1 , … , x i + k , x i + 1 , x i , … , x i + k − 2 , x i + k − 1 , … , x n ) ( − 1 ) k + ( k − 1 ) ϕ ( x 1 , … , x i + k , … , x i + k − 1 , x i , … , x n ) − ϕ ( x 1 , … , x i + k , … , x i + k − 1 , x i , … , x n )
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Proof (2) ϕ ( x 1 , … , x i , x i , … , x n ) = − ϕ ( x 1 , … , x i , x i , … , x n ) ⟺ 2 ϕ ( x 1 , … , x i , x i , … , x n ) = 0 ⟺ ϕ ( x 1 , … , x i , x i , … , x n ) = 0
\begin{align*}
&& \phi (x_{1}, \dots, x_{i}, x_{i}, \dots, x_{n}) =&\ - \phi (x_{1}, \dots, x_{i}, x_{i}, \dots, x_{n})
\\ \iff && 2\phi (x_{1}, \dots, x_{i}, x_{i}, \dots, x_{n}) =&\ 0
\\ \iff && \phi (x_{1}, \dots, x_{i}, x_{i}, \dots, x_{n}) =&\ 0
\end{align*}
⟺ ⟺ ϕ ( x 1 , … , x i , x i , … , x n ) = 2 ϕ ( x 1 , … , x i , x i , … , x n ) = ϕ ( x 1 , … , x i , x i , … , x n ) = − ϕ ( x 1 , … , x i , x i , … , x n ) 0 0
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