Exchange of Order of Summation with Dependency on Both Indices/Proof
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Theorem
- $\ds \sum_{\map R i} \sum_{\map S {i, j} } a_{i j} = \sum_{\map {S'} j} \sum_{\map {R'} {i, j} } a_{i j}$
where:
- $\map {S'} j$ denotes the propositional function:
- there exists an $i$ such that both $\map R i$ and $\map S {i, j}$ hold
- $\map {R'} {i, j}$ denotes the propositional function:
- both $\map R i$ and $\map S {i, j}$ hold.
Proof
\(\ds \sum_{\map R i} \sum_{\map S {i, j} } a_{i j}\) | \(=\) | \(\ds \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i} \sqbrk {\map S {i, j} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i \land \map S {i, j} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map {S'} j} \sqbrk {\map {R'} {i, j} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{\map {S'} j} \sum_{\map {R'} {i, j} } a_{i j}\) |
$\blacksquare$
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: $(21)$