Example 2 The number of sets $A$ that satisfy $\{a, b\} \subseteq A \varsubsetneqq \{a, b, c, d\}$ is $\qquad$.

The number of sets $A$ that satisfy the condition is $2^{2}-1=3$.
Key to the solution The essence of this example is to find the number of proper subsets of the set $\{c, d\}$; generally, if $n, k \in \mathbf{N}$, $k<n$, then the number of sets $A$ that satisfy $\left\{a_{1}, a_{2}, \cdots, a_{k}\right\} \subseteq A \varsubsetneqq\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ is $2^{n-k}-1$.