Question

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

Answer 1

$$\begin{gathered} \mathrm{LHS}=\left(A\cup B\right)\cup \left(A\cap B\text{'}\right) \\ \Rightarrow \mathrm{LHS}=\left\{\left(A\cup B\right)\cap A\right\}\cup \left\{\left(A\cup B\right)\cap B\text{'}\right\} \\ \Rightarrow \mathrm{LHS}=\left\{\left(A\cup B\right)\cap A\right\}\cup \left\{\left(A\cup B\right)\cap B\text{'}\right\} \\ \Rightarrow \mathrm{LHS}=A\cup \left\{\left(A\cup B\right)\cap B\text{'}\right\} \\ \Rightarrow \mathrm{LHS}=A\cup \left\{\left(A\cap B\text{'}\right)\cup \left(B\cap B\text{'}\right)\right\}\left(\because B\cap B=\varphi \right) \\ \Rightarrow \mathrm{LHS}=A\cup \left(A\cap B\text{'}\right) \\ \Rightarrow \mathrm{LHS}=A=\mathrm{RHS} \end{gathered}$$