Question

If $A,B$ and $C$are three sets such that $A\cap B=A\cap C$and $A\cup B=A\cup C$, then

• A. $A=C$
• B. $B=C$
• C. $A\cap B=\varphi$
• D. $A=B$

Answer 1

The correct option is B

$B=C$

Finding the value:

Given that $A\cup B=A\cup C$

$$\begin{gathered} \Rightarrow (A\cup B)\cap C=(A\cup C)\cap C \\ \Rightarrow (A\cap C)\cup (B\cap C)=C\left[\right(A\cup C)\cap C=C] \\ \Rightarrow (A\cap B)U(B\cap C)=C\dots \dots \dots .\left(1\right)\mathbf{}\mathbf{}\mathbf{\left[}\mathbf{\right(}\mathit{A}\mathbf{}\mathbf{\cap }\mathbf{}\mathit{C}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{A}\mathbf{}\mathbf{\cap }\mathbf{}\mathit{B}\mathbf{}\mathbf{]} \end{gathered}$$

Take $A\cup B=A\cup C$

$$\begin{gathered} \Rightarrow (A\cup B)\cap B=(A\cup C)\cap B \\ \Rightarrow B=(A\cap B)\cup (C\cap B) \\ \Rightarrow (A\cap B)\cup (B\cap C)=B\dots \dots \dots ..\left(2\right) \end{gathered}$$

From equations 1 & 2, we get

Therefore, $B=C$

Hence the correct option is B.