Prove that A is a subset of B iff B' is subset of A'
i would prove both direction via a direct proof
so to start you off
(=>) Assume $\displaystyle A \subseteq B$. Then, $\displaystyle x \in A \implies x \in B$. Now, let $\displaystyle y \in B'$ .....(you want to show that this implies $\displaystyle y \in A'$)
(<=) For the converse, assume $\displaystyle B' \subseteq A'$. Then $\displaystyle x \in B' \implies x \in A'$. Now, let $\displaystyle y \in A$.....(you want to show that this implies $\displaystyle y \in B$)
now continue