Change of Variables in Summation over Finite Set

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

Let $S$ and $T$ be finite sets.

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

Let $g: T\to S$ be a bijection.

Then we have an equality of summations over finite sets:


 * $\displaystyle \sum_{s \mathop \in S}f(s) = \sum_{t \mathop \in T} f(g(t))$

Outline of Proof
This follows from the definition of summation over finite set and the fact that Summation over Finite Set is Well-Defined.

Proof
Let $n$ be the cardinality of $S$ and $T$.

Let $\N_{<n}$ be an initial segment of the natural numbers.

Let $h : \N_{<n} \to T$ be a bijection.

By definition of summation over finite set, $\displaystyle \sum_{t \mathop\in T} f(g(t)) = \displaystyle \sum_{i\mathop= 0}^{n-1} f(g(h(i)))$.

By Composite of Bijections is Bijection, the composition $g\circ h : \N_{<n} \to S$ is a bijection.

By definition of summation over finite set, $\displaystyle \sum_{s \mathop\in S} f(s) = \displaystyle \sum_{i\mathop= 0}^{n-1} f(g(h(i)))$.

Also see

 * Change of Variables in Indexed Summation
 * Finite Summation does not Change under Permutation