Formal Definition of a Finite Set

\(S \sim \{1, 2, 3, \ldots, n\}\) for some \(n \in \mathbb{N}\)

Explanation

The formal definition: \(S\) is finite with \(n\) elements if \(S \sim \{1, 2, 3, \ldots, n\}\) — i.e., a bijection exists between \(S\) and the set of the first \(n\) natural numbers. The empty set is considered finite (it corresponds to \(n = 0\)). Two finite sets are equivalent if and only if they have the same number of elements.