Is this set a contradiction or just confusing? Book says "Prove that:$$A\cap B^c\subset B\Leftrightarrow A\subset B"$$
But if $A\subset B $ then $A\cap B^c=\emptyset$ isn't it?
is book wrong?

Suppose $A\cap B^c\subseteq B$. Suppose there's a $b\in A\cap B^c$, then $b\in A$ and $b\not\in B$, but from hypothesis, we have that $b\in B$, a contradiction, thus $A\cap B^c$ is empty.
Now, the set $A\cap B^c$ consists of all the elements that are in $A$ and in $B^c$, this means that (as the intersection is empty), if $a$ is $A$, then $a$ is not in $B^c$, that is, $a\in B$.
