# Exchange of Order of Product

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## Contents

## Theorem

Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ and $S: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be propositional functions on the set of integers.

Let $\displaystyle \prod_{R \left({i}\right)} x_i$ denote a product over $R$.

Let the fiber of truth of both $R$ and $S$ be finite.

Then:

- $\displaystyle \prod_{R \left({i}\right)} \prod_{S \left({j}\right)} a_{i j} = \prod_{S \left({j}\right)} \prod_{R \left({i}\right)} a_{i j}$

## Proof

## Also known as

The word **interchange** can often be seen for **exchange**.

## 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