A is a proper subset of B if A is a subset of B and A isn't equal to B. In other words, A has some but not all of the elements of B, and A doesn't have any elements that don't belong to B.

We can also say that B is a proper superset of A.

Let A and B be two sets. We say that A is a subset of B if every element of A is also an element of B. In other words, A consists of some (possibly all) of the elements of B but doesn't have any elements that B doesn't have. If A is a subset of B, we can also say that B is a superset of A.

Given a set A with cardinality n, there are 2ⁿ subsets, and 2ⁿ − 1 proper subsets. Here's why:

1. Consider a set with one element, {a}. There are two subsets: ∅ and {a}.
2. Now consider a set with two elements, {a, b}. There are four subsets: ∅, {a}, {b}, and {a, b}.
3. This pattern goes on for any n, to deliver 2ⁿ.
4. However, for the number of proper subsets, we subtract one subset (representing the original set, A): 2ⁿ − 1.