If A, B, C be three sets such that $A∪B = A∪C$ and $A∩B=A∩C$, then
Answer & explanation
Correct answer: option 2
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 $B = C$.