Using properties of sets, show that for any two sets A and B, (A … - Vidyadip

Question

Using properties of sets, show that for any two sets A and $\text{B},\text{ (A}\cup\text{B})\cap(\text{A}\cup\text{B}')=\text{A}.$

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Answer

We need to show $(\text{A}\cup\text{B})\cap(\text{A}\cap\text{B}')=\text{A}$
Now,
$(\text{A}\cup\text{B})\cap(\text{A}\cap\text{B}')=((\text{A}\cup\text{B})\cap\text{A})\cap\text{B}'$
$=((\text{A}\cap\text{A})\cup(\text{B}\cap\text{A}))\cap\text{B}'$
$=\text{A}\cap\text{B}'$
$=\text{A}.$

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