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

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Solution

We need to show $(A \cup B) \cap (A \cap B)^{'} = A$

Now,

$(A \cup B) \cap (A \cap B^{'})$ = { $(A \cup B) \cap A$ } $\cap B^{'}$

[Using associative property]

= { $(A \cap A) \cup (B \cap A)$ } $\cap B^{'}$

[ $∴ A \cap A = A a n d B \cap A = A \cap B,$ by commutative law]

= $A \cap B^{'}$ [ $∵ A \cup (A \cap B) = A$ ]

= A

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