Formal Definition of Summation over a Set

Assume that $\lvert S \rvert = n$. Then there is a bijection between the set $\left\{1,2,\dots,n\right\}$ and $S$. That is, there exists a mapping $f: \left\{1,2,\dots,n\right\} \mapsto S$ and the inverse $f^{-1}: S \mapsto \left\{1,2,\dots,n\right\}$ exists. Then we may write $$\sum_{s \in S}h_{s} = \sum_{i=1}^{n}h_{f_{i}},$$ where $h: S \mapsto \mathbb{R}$ is the sequence we'd like to sum. This definition incorporates the definition of determinants, as a bijection between permutations of the sequence $\left\{1,2,\dots,n\right\}$ and a subset of natural numbers always exists.