Question

Prove that AUB = A∩B => A=B

Answer 1

`\sf\\\textsf{Let }x \in A. \textsf{ Then, x}\in A \cup B.\\\textsf{As }A\cup B=A\cap B,\\\textsf{ \ \ }x\in A\cap B\\ So,\ x\in B.\\\textsf{i.e. if an element belongs to A, it must belong to B too.}\\\therefore A \subset B......(1)`

`\sf\\\textsf{Now let y}\in B.\ Then, y\in A\cup B.\\\textsf{Since }A\cup B=A\cap B,\\y\in A\cap B\\ So,\ y\in A.\\\textsf{i.e. if an element belongs to B, it must belong to A also.}\\\therefore\ B \subset A.......(2)`

`\sf\\\textsf{From (1) and (2),}\\(A\subset B)\textsf{ and }(B\subset A)\\\therefore\ A=B\ \ \ \ \ \ proved`