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

Question

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

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Answer

$\text{LHS}=\text{A}\cap\text{(A}\cup\text{B}')$ $=\text{A}\cap\text{(A}'\cap\text{B}')$ [By De-morgan's law] $=\text{(A}\cap\text{A}')\cap\text{B}'$ [By associative law] $= \oint \cap \text{ B}'$ $[\because \text{A}\cap{A}' =\oint]$ $= \oint$ $= \text{RHS}$ $\therefore$ LHS = RHS Proved.

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