# Using properties of sets, show that for any two sets A and B,

Question:

Using properties of sets, show that for any two sets A and B,

$(A \cup B) \cap\left(A \cap B^{\prime}\right)=A$

Solution:

LHS $=(A \cup B) \cup\left(A \cap B^{\prime}\right)$

$\Rightarrow \mathrm{LHS}=\{(A \cup B) \cap A\} \cup\left\{(A \cup B) \cap B^{\prime}\right\}$

$\Rightarrow \mathrm{LHS}=\{(A \cup B) \cap A\} \cup\left\{(A \cup B) \cap B^{\prime}\right\}$

$\Rightarrow \mathrm{LHS}=A \cup\left\{(A \cup B) \cap B^{\prime}\right\}$

$\Rightarrow \mathrm{LHS}=A \cup\left\{\left(A \cap B^{\prime}\right) \cup\left(B \cap B^{\prime}\right)\right\} \quad(\because B \cap B=\phi)$

$\Rightarrow \mathrm{LHS}=A \cup\left(A \cap B^{\prime}\right)$

$\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$