Example 1.4 Find the number of subsets of the $n$-element set $A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$.
Solution: We can construct a subset of an $n$-element set $A$ through $n$ steps, where the $k$-th step $(k=1,2, \cdots, n)$ is to determine whether to select $a_{k}$ as an element of the subset. Since there are two ways to complete each step, by the multiplication principle, the number of ways to construct a subset of the $n$-element set $A$ is $2^{n}$, meaning that the $n$-element set $A$ has $2^{n}$ subsets.