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\\ot\in A`, so it's not always the case that
`a\in A\cap B`, and therefore in general
`A\cup B\\ot\subseteq A\cap B`.