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:
- $\ds \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$
where:
- $\ds \sum_{\map R j} a_j$ 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 $\ds \sum_{\map R i} a_i$ be absolutely convergent.
Then:
- $\ds \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$
Proof
\(\ds \sum_{\map R {\map \pi j} } a_{\map \pi j}\) | \(=\) | \(\ds \sum_{j \mathop \in \Z} a_{\map \pi j} \sqbrk {\map R {\map \pi j} }\) | Definition of Summation by Iverson's Convention | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in \Z} \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i} \sqbrk {i = \map \pi j}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i} \sum_{j \mathop \in \Z} \sqbrk {i = \map \pi j}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{\map R i} a_i\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{\map R j} a_j\) | 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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: $(5)$