Question

If $(A\cup B)=(A\cap B)$ then prove that A=B

Answer 1

Let $(A\cup B)=(A\cap B)$ be given.

Let x be an arbitrary element of A. Then,

$\begin{array}{cc}xϵAandxϵB\Rightarrow xϵA\cup B& [\because A\subseteq A\cup B]\\ \Rightarrow xϵA\cap B& [\because A\cup B=A\cap B]\\ \Rightarrow xϵAandxϵB& \\ \Rightarrow xϵB\text{(}surely).& \\ \therefore A\subseteq B& \dots \left(i\right)\end{array}$

Again, let $yϵB$. Then,

$\begin{array}{ccc}yϵB& \Rightarrow yϵA\cup B& [\because B\subseteq A\cup B]\\ & \Rightarrow yϵA\cap B& \because A\cup B=A\cap B\\ & \Rightarrow yϵAand\Rightarrow yϵB& \\ & yϵA\left(surely\right).& \\ \therefore & B\subseteq A.& \dots \left(ii\right)\end{array}$

Thus from (i) and (ii), we get A= B