Let A and B be two sets that $A∩X=B∩X=\phi$ and $A∪X=B∪X$ for some set X. Then,

$A = B$ $A = X$ $B = X$ $A ∪ B = X$

$A = B$

The correct answer is Option (1) → $A = B$ We have, $A∪X=B∪X$ for some set X $⇒A∩(A∪X)=A∩(B∪X)$ $⇒A=(A∩B)∪(A∩X)$  $[∵ A∩(A∪X) = A]$ $⇒A = (A∩B)∪\phi$   $[∵ A∩X=\phi]$ $⇒A = A∩B$  ...(i) Again $A∪X=B∪X$ $⇒B∪(A∪X)=B∩(B∪X)$ $⇒ (B∩A)∪(B∩X) = B$ $⇒(B∩A)∪\phi=B$ $⇒A∩B=B$  ...(ii) From (i) and (ii), we have $A = B$