Answer
a)-d) See proofs
Work Step by Step
a) We have to prove:
$(A\cap B)\subseteq A$
Consider $x\in (A\cap B)$.
We have:
$x\in A$ and $A\in A$.
We got:
$(A\cap B)\subseteq A$\\
b) We have to prove:
$A\subseteq (A\cup B)$
Consider $x\in A$.
We have:
$x\in A\Rightarrow x\in A$ or $x\in B\Rightarrow x\in (A\cup B)$.
We got:
$A\subseteq (A\cup B)$
c) We have to prove:
$A-B\subseteq A$
Consider $x\in A-B$.
We have:
$x\in A$ and $x\not in B$
We got:\\
$A-B\subseteq A$
d) We have to prove:
$A\cap(B-A)=\phi$
Consider $x\in A$.
We have:
$x\in A\cap(B-A)$.
We have:
$x\in A$ and $x\in (B-A)$
$x\in A$ and ($x\in B$ and $x\not in A$)
We got $x\in A$ and $x\not in A$.
As this is not possible, we have:\\
$A\cap(B-A)=\phi$
