> [!NOTE] Definition (Cardinality) > Suppose $X$ and $Y$ are [[Sets|sets]]. If there is some [[Bijection|bijection]] $f:X \to Y$ then we say that the sets have the same ***cardinality***. # Properties **Cardinality of finite sets**: A set $S$ is [[Finite Set|finite]] if there is a bijection between it and $\{ i \in \mathbb{N} \mid i< n \}$ for some $n \in \mathbb{N}.$ In this case we write $|S|=n.$ Note also that it has [[Uniqueness of cardinality of finite sets|unique cardinality]]. # Applications **Examples**: See [[Cardinality of Power Set of Finite Set]].