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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