Change of Index Variable of Product

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Theorem

$\displaystyle \prod_{R \left({i}\right)} a_i = \prod_{R \left({j}\right)} a_j$

where $\displaystyle \prod_{R \left({i}\right)} a_i$ denotes the product over $a_i$ for all $i$ that satisfy the propositional function $R \left({i}\right)$.


Proof

Let $S = \left\{ {i \in \Z: R \left({i}\right)}\right\}$.

Let $T = \left\{ {j \in \Z: R \left({j}\right)}\right\}$.

Let $i \in S$.

Then $R \left({i}\right)$.

Let $j = i$.

By Leibniz's Law, $R \left({j}\right)$.

Thus $i \in T$.

By definition of subset, $S \subseteq T$.


Similarly, let $j \in T$.

Then $R \left({j}\right)$.

Let $i = j$.

By Leibniz's Law, $R \left({i}\right)$.

Thus $j \in S$.

By definition of subset, $T \subseteq S$.


Thus by definition of set equality: $S = T$


Thus:

\(\displaystyle \prod_{R \left({i}\right)} a_i\) \(=\) \(\displaystyle \prod a_i \left[{R \left({i}\right)}\right]\) $\quad$ Definition of Iverson's Convention $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \prod a_i \times \chi_S\) $\quad$ Definition of Characteristic Function of Set $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \prod a_j \times \chi_T\) $\quad$ as $S = T$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \prod a_j \left[{R \left({j}\right)}\right]\) $\quad$ Definition of Iverson's Convention $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \prod_{R \left({j}\right)} a_j\) $\quad$ $\quad$

$\blacksquare$


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