Exchange of Order of Summations over Finite Sets

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Theorem

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.

Let $S, T$ be finite sets.

Let $S \times T$ be their cartesian product.


Cartesian Product

Let $f: S \times T \to \mathbb A$ be a mapping.


Then we have an equality of summations over finite sets:

$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in T} f \left({s, t}\right) = \sum_{t \mathop \in T} \sum_{s \mathop \in S} f \left({s, t}\right)$


Subset of Cartesian Product

Let $D\subset S \times T$ be a subset.

Let $\pi_1 : D \to S$ and $\pi_2 : D \to T$ be the restrictions of the projections of $S\times T$.


Then we have an equality of summations over finite sets:

$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in \pi_2 \left({\pi_1^{-1} \left({s}\right)}\right)} f \left({s, t}\right) = \sum_{t \mathop \in T} \sum_{s \mathop \in \pi_1 \left({\pi_2^{-1} \left({t}\right)}\right)} f \left({s, t}\right)$


Also see