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
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,
$A∪B=A∪C$
$⇒(A∪B)∩B=(A∪C)∩B$
$⇒B=(A∩B)∪(B∩C)$ ...(ii)
From (i) and (ii), we get $B = C$.