Cardinality — Counting the Elements of a Set

The cardinality of a finite set $A$ is the number of distinct elements it contains, written $n(A)$ or $|A|$. For example, if $A = \{3, 6, 9, 12\}$, then $n(A) = 4$.

A few immediate facts follow directly from the definition:

• $n(\emptyset) = 0$ — the empty set has no elements.
• If $A$ is a singleton set, $n(A) = 1$.
• If $A$ and $B$ are finite sets, the cardinality of their Cartesian product is

This is simply because every one of $A$'s $n(A)$ elements is paired with every one of $B$'s $n(B)$ elements exactly once when forming ordered pairs, giving $n(A) \times n(B)$ pairs in total. …