For any two sets A and B, prove the following: A (A' )=A - Vidyadip

Question

For any two sets A and B, prove the following:
$\text{A}\cap\text{(A}'\cup\text{B})=\text{A}\cap\text{B}$

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Answer

$\text{LHS}=\text{A}\cap\text{(A}'\cup\text{B)}$
$=\text{(A}\cap\text{A}')\cup\text{(A}\cap\text{B)}$ $[\because\cap$ distributes over (i)$]$
$= \oint\cup\text{ (A}\cap\text{B)}$ $[\because\text{A}\cap\text{A}' = \oint]$
$=\text{A}\cap\text{B}$ $[\because\oint\cup$ x = x for any set x$]$
$= \text{RHS}$
$\therefore$ LHS = RHS Proved.

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