Question

If A, B and C be the sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C$ then prove that B= C

Answer 1

Let $A\cup B=A\cup C$ and $A\cap B=A\cap C$ be given Then.
$A\cup B=A\cup C$

$\Rightarrow (A\cup B)\cap B=(A\cup C)\cap Band(A\cup B)\cap C=(A\cup C)\cap C$

$\Rightarrow B=(A\cap B])\cup (C\cap B\left)and\right(A\cap C)\cup (B\cap C)=C$

$[\because B\subseteq (A\cup B)andC\subseteq (A\cup C\left)\right]$

$\Rightarrow B=(A\cap B)\cup (B\cap C)and(A\cap B)\cup (B\cap C)=C[\because A\cap C=A\cap B]$

$\Rightarrow B=C$

Hence, B=C