Question

Show that for any sets A and B, ( By using the properties of sets )
$A=\left(A\cap B\right)\cap \left(A-B\right)$

Answer 1

To Proof:-$A=(A\cap B)\cup (A-B)$

$A-B=$Eleement of $A$ that is not in $B$.

$A\cap B=$Element which are in both $A$ & $B$.

$\because A-B=A\cap B^{{}^{\prime }}(B^{{}^{\prime }}$ is complement of $B)$

$\therefore (A\cap B)\cup (A\cap B^{{}^{\prime }})$

$=A\cap (B\cup B^{{}^{\prime }})$[Distributive property of sets]

$=A\cap \cup [B\cup B^{{}^{\prime }}=\cup ]=A$