Question

Let $A,B$ and $C$ be the sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C.$ Show that $B=C$

Answer 1

Given that AUB = AUC
⇒ (AUB) ∩ C = (AUC) ∩C
⇒ (A∩C) U (B∩C) = C [ ∴(AUC)∩C = C ]
⇒ (A∩B) U (B∩C) = C ..........(1) [ ∴(A∩C) = A∩B ]
Again AUB = AUC
(AUB) ∩ B = (AUC) ∩ B
B = (A∩B) U (C∩B)
= (A∩B) U (B∩C) ...........(2)
From 1 & 2 we get
B = C