= Solution
Complements are relative to the same <universal set> $\mathcal U$. Suppose $A\subseteq B$ and $x\in\overline B$. Then $x\in\mathcal U$ and $x\notin B$. If $x$ belonged to $A$, the assumed <set inclusion> would imply $x\in B$, a contradiction. Thus $x\notin A$ and $x\in\overline A$, proving
$$
\boxed{A\subseteq B\ \Longrightarrow\ \overline B\subseteq\overline A.}
$$
There is only one <set inclusion> in this conclusion; a reverse inclusion is neither needed nor generally true. The request for double inclusion applies to the equalities in (i) and (ii).
Back to article page
