For any two sets, prove that: A (A )=A. - Vidyadip

Question

For any two sets, prove that:
$\text{A}\cup(\text{A}\cap\text{B})=\text{A}.$

✓

Answer

$\text{A}\cup(\text{A}\cap\text{B})=(\text{A}\cup\text{A})\cap(\text{A}\cup\text{B})$ $[\because$ union $\cup$ is distributive over intersection $\cap]$
$=\text{A}\cap(\text{A}\cup\text{B})$ $[\because\text{A}\cup\text{A}=\text{A}]$
$=\text{A}[\because\text{A}\subset(\text{A}\cup\text{B}),$ as union of two sets is bigger then each of the individual sets$]$
Hence, $\text{A}\cup(\text{A}\cap\text{B})=\text{A}$ Proved.

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