509: For Set of Sequences for Fixed Domain and Codomain, Permutation Bijectively Maps Set onto Set

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description/proof of that for set of sequences for fixed domain and codomain, permutation bijectively maps set onto set

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof
• 4: Note

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Starting Context

• The reader knows a definition of permutation of sequence.

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Target Context

• The reader will have a description and a proof of the proposition that for the set of all the sequences for any fixed domain and any fixed codomain, any permutation bijectively maps the set onto the set.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$S$: $\subseteq \mathbb{N}$, $=\{l_1,...,l_k\}$
$S^{\prime }$: $\in \{\text{ the sets }\}$
$S^{″}$: $=\{f|f:S\to S^{\prime }\in \{\text{ the sequences over }S\}\}$
$\sigma$: $\in \{\text{ the permutations of an }{f}_0\in S^{″}\}$, $:S\to S$
$f_{\sigma }$: $S^{″}\to S^{″},f\mapsto f\circ \sigma$
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Statements:
$f_{\sigma }\in \{\text{ the bijections }\}$.
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2: Natural Language Description

For any subset, $S\subseteq \mathbb{N}=\{l_1,...,l_k\}$, any set, $S^{\prime }$, the set of all the sequences, $S^{″}=\{f|f:S\to S^{\prime }\}$, and any permutation, $\sigma :S\to S$ of any $f_0\in S^{″}$, $f_{\sigma }:S^{″}\to S^{″},f\mapsto f\circ \sigma$ is a bijection.

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3: Proof

$f_{\sigma }$ is well-defined, because $f\circ \sigma :S\to S\to S^{\prime }\in S^{″}$.

Let us prove that $f_{\sigma }$ is injective. For any $f\ne f^{\prime }\in S^{″}$, $f\circ \sigma \ne f^{\prime }\circ \sigma$, because if $f\circ \sigma =f^{\prime }\circ \sigma$, $f=f\circ \sigma \circ {\sigma }^{-1}=f^{\prime }\circ \sigma \circ {\sigma }^{-1}=f^{\prime }$, a contradiction.

Let us prove that $f_{\sigma }$ is surjective. For any $f^{\prime }\in S^{″}$, $f^{\prime }\circ {\sigma }^{-1}\in S^{″}$ and $f^{\prime }\circ {\sigma }^{-1}\circ \sigma =f^{\prime }$.

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4: Note

Typically, $S=\{1,...,k\}$ and $S^{\prime }=\{1,...,d\}$ and we think of $\sum _{(j_1,...,j_k)\in S^{″}}{N}^{j_1,...,j_k}{M}_{j_1,...,j_k}$, which is nothing but $N^{j_1,...,j_k}{M}_{j_1,...,j_k}$ with the Einstein convention. What this proposition implies is that $\sum _{(j_1,...,j_k)\in S^{″}}{N}^{\sigma ((j_1,...,j_k){)}_1,...,\sigma ((j_1,...,j_k){)}_k}{M}_{\sigma ((j_1,...,j_k){)}_1,...,\sigma ((j_1,...,j_k){)}_k}=N^{j_1,...,j_k}{M}_{j_1,...,j_k}$. That fact is usually immediately accepted as obvious, but we have bothered to prove it rigorously.

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References

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