If A, B, C be three sets such that $A∪B = A∪C$ and $A∩B=A∩C$, then

$A = B$ $B = C$ $A = C$ $A = B = C$

$B = C$

The correct answer is Option (2) → $B = C$ We have, $A∪B = A∪C$ $⇒(A∪B)∩C=(A∪C)∩C$ $⇒(A∩C)∪(B∩C)=C$ $⇒(A∩B)∪(B∩C)=C$  $[∵ A∩C=A∩B]$ .....(i) Again, $A∪B = A∪C$ $⇒(A∪B)∩B=(A∪C)∩B$ $⇒B=(A∩B)∪(B∩C)$ From (i) and (ii), we get $B = C$.