Cardinality of sets

Equal cardinality

We say that two sets $X$ and $Y$ have equal cardinality iff there exists a bijection $f:X\to Y$ from $X$ to $Y$.

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 $\{i\in \mathbb{N}:1\ge i\ge n\}$. We also say that such a set has $n$ elements.

Finite sets

A set is finite iff it has cardinality $n$ for some natural number $n$; otherwise, the set is called infinite.

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