Let A and B be two sets that $A∩X=B∩X=\phi$ and $A∪X=B∪X$ for some set X. Then,
Answer & explanation
Correct answer: option 1
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
$A = B$