Question

If A ⊂ B, then A ∩ B = A ∪ B. always sometimes or never

Answer 1

Sometimes. Proof:

Assume $A\subset B$.

Let $a\in A\cap B$, so that $a\in A$ and $a\in B$. Then of course $a\in A\cup B$, so $A\cap B\subseteq A\cup B$.

Now let $a\in A\cup B$, so that $a\in A$ *or* $a\in B$. Since $A\subset B$, if $a\in A$ then $a\in B$, so that $a\in A\cap B$. But if $a\in B$, then $a\not\in A$, so it's not always the case that $a\in A\cap B$, and therefore in general $A\cup B\not\subseteq A\cap B$.