Using properties of sets, show that for any two sets A and B,
$(A \cup B) \cap\left(A \cap B^{\prime}\right)=A$
LHS $=(A \cup B) \cup\left(A \cap B^{\prime}\right)$
$\Rightarrow \mathrm{LHS}=\{(A \cup B) \cap A\} \cup\left\{(A \cup B) \cap B^{\prime}\right\}$
$\Rightarrow \mathrm{LHS}=\{(A \cup B) \cap A\} \cup\left\{(A \cup B) \cap B^{\prime}\right\}$
$\Rightarrow \mathrm{LHS}=A \cup\left\{(A \cup B) \cap B^{\prime}\right\}$
$\Rightarrow \mathrm{LHS}=A \cup\left\{\left(A \cap B^{\prime}\right) \cup\left(B \cap B^{\prime}\right)\right\} \quad(\because B \cap B=\phi)$
$\Rightarrow \mathrm{LHS}=A \cup\left(A \cap B^{\prime}\right)$
$\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$
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