The most straightforward way is to use subsets of the integers (or rationals). For (a), let $A=\mathbb{N}\cup \\{-1\\}$ and let $B=\mathbb{N}\cup \\{-2\\}$. Then their intersection is denumerable. Construction like this, where their intersection is a set well known to be denumerable, is the easiest way to go. For (b), try choosing a set $A$ that mostly contains $B$. In other words, add an extra element to $B$ so that $B$ is not a subset of $A$. Make sure that the compliment of $B$ in $A$ is also denumberable. The even/odd numbers in $\mathbb{N}$ should work well for this.