Question

Let A and B be sets. Prove that (a) ACB AUB=B; (b) ACB = AnB = A.

Answer 1

For part (a) and (b) suposse
`A\subset B`.

To prove part (a) observe that we already have that
`B\subset A\cup B`. So we will prove that
`A\cup B \subset B`. Let
`x\in A\cup B`, then
`x\in A` or
`x\in B`. If
`x\in B` we finish the proof, and if
`x\in A` implies
`x\in B` because we assume
`A\subset B`, and the proof is complete.

For part (b) we always have
`A\cap B\subset A`. We finish the proof showing
`A\subset A\cap B`. Let
`x\in A`, then
`x\in B` by the asumption that
`A\subset B`. So, we have both
`x\in A` and
`x\in B`, that implies
`x\in A\cap B`. Therfore
`A\subset A\cap B`, which completes the proof.