Permutation of Indices of Summation

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Theorem

Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers.

Let the fiber of truth of $R$ be finite.


Then:

$\displaystyle \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$

where:

$\displaystyle \sum_{\map R j} a_j$ denotes the summation over $a_j$ for all $j$ that satisfy the propositional function $\map R j$
$\pi$ is a permutation on the fiber of truth of $R$.


Infinite Series

Let the fiber of truth of $R$ be infinite.


Let $\displaystyle \sum_{\map R i} a_i$ be absolutely convergent.


Then:

$\displaystyle \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$


Proof

\(\displaystyle \sum_{R \left({\pi \left({j}\right)}\right)} a_{\pi \left({j}\right)}\) \(=\) \(\displaystyle \sum_{j \mathop \in \Z} a_{\pi \left({j}\right)} \left[{R \left({\pi \left({j}\right)}\right)}\right]\) Definition of Summation by Iverson's Convention
\(\displaystyle \) \(=\) \(\displaystyle \sum_{j \mathop \in \Z} \sum_{i \mathop \in \Z} a_i \left[{R \left({i}\right)}\right] \left[{i = \pi \left({j}\right)}\right]\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop \in \Z} a_i \left[{R \left({i}\right)}\right] \sum_{j \mathop \in \Z} \left[{i = \pi \left({j}\right)}\right]\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop \in \Z} a_i \left[{R \left({i}\right)}\right]\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{R \left({i}\right)} a_i\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{R \left({j}\right)} a_j\) Change of Index Variable of Summation

$\blacksquare$


Also known as

The operation of permutation of indices of a summation can be seen referred to as a permutation of the range.

However, as the term range is ambiguous in the literature, and as its use here is not strictly accurate (it is the fiber of truth of $R$, not its range, which is being permuted, its use on $\mathsf{Pr} \infty \mathsf{fWiki}$ is discouraged.


Sources