Question

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

• A. $A=B$
• B. $B=C$
• C. $A=C$
• D. $A=B=C$

Answer 1

The correct option is B

$B=C$

Finding the value:

$\begin{array}{ccc}A\cup B& =A\cup C& \\ \Rightarrow (A\cup B)\cap C& =(A\cup C)\cap C& \\ \Rightarrow (A\cap C)\cup (B\cap C)& =C& \\ \Rightarrow (A\cap B)\cup (B\cap C)& =C....\left(1\right)(\because A\cap C=A\cap B)& \\ \Rightarrow A\cup B& =A\cup C& \\ \Rightarrow (A\cup B)\cap B& =(A\cup C)\cap B& \\ \Rightarrow B& =(A\cap B)\cup (C\cap B)& \\ & =(A\cap B)\cup (B\cap C)....\left(2\right)& \end{array}$

From $\left(1\right)$ and $\left(2\right)$

$B=C$

Explanation for the incorrect options:

Option $\left(A\right)$: $A=B$

Since $A\cup B=A\cup C$ and $A\cap B=A\cap C$ $⟹B=C$

Option $\left(C\right)$: $A=C$

Since $A\cup B=A\cup C$ and $A\cap B=A\cap C$ $⟹B=C$

Option $\left(D\right)$: $A=B=C$

Since $A\cup B=A\cup C$ and $A\cap B=A\cap C$ $⟹B=C$

Hence option $\left(B\right)$ is the correct option.