\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)

Cardinality of sets

Equal cardinality

We say that two sets \( X \) and \( Y \) have equal cardinality iff [...].

Cardinality n

Let \( n \) be a natural number. A set \( X \) is said to have cardinality n if it has equal cardinality with the set [...]. We also say that such a set has \( n \) elements.

Finite sets

A set is finite iff [...] for some natural number \( n \); otherwise, the set is called infinite.

Notation: if \( X \) is a finite set, we use \( \#(X) \) to denote the cardinality of \( X \).