So I've come across this interesting proof question in my discrete mathematics textbook that I'm trying to solve for practice but unfortunately it does not have a solution. Any help with a reasonable solution and explanation so that I can verify my work would be greatly appreciated:
Suppose that $U$ is the universal set, and that $A$ and $B$ are two arbitrary sets of elements of $U$:
a. First, suppose that $A$ contains at least two elements. Using a direct proof, show that if every proper subset of $A$ is a subset of $B$, then $A$ is a subset of $B$.
b. Give an example that shows that the implication from part (a) is False if $A$ contains only one element.