Question

Using properties of sets show that:
$\left(i\right)A\cup (A\cap B)=A$
$\left(ii\right)A\cap (A\cup B)=A$

Answer 1

$\left(i\right)A\cup (A\cap B)=A$
L.H.S $=A\cup (A\cap B)$
$=(A\cup A)\cap (A\cup B)\left\{\text{Using distributive law}\right\}$
$=A\cap (A\cup B)$
$=A$
= R.H.S
Thus, $A\cup (A\cap B)=A$
Hence proved.

$\left(ii\right)A\cap (A\cup B)=A$
L.H.S. $=A\cap (A\cup B)$
$=(A\cap A)\cup (A\cap B)\left\{\text{Using distributive law}\right\}$
$=A\cup (A\cap B)$
$=A$
= R.H.S
Thus, $A\cap (A\cup B)=A$
Hence proved.