Question

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

Answer 1

(i) To show: $A\cup (A\cap B)=A$
We know that
$A\subset A$
$\Rightarrow A\cap B\subset A$
$\therefore A\cup (A\cap B)\subset A$ ......(1)
Also, $A\subset A\cup (A\cap B)$ ............(2)
$\therefore$ From (1) and (2),
$A\cup (A\cap B)=A$
(ii) To show: $A\cap (A\cup B)=A$
$A\cap (A\cup B)=(A\cap A)\cup (A\cap B)$
$=A\cup (A\cap B)$
$=A$ (by (i))
$\Rightarrow A\cap (A\cup B)=A$