Question

Three sets $A,B$ and $C$ are such that $A=B\cap C$ and $B=C\cap A$, then

• A. $A\subset B$
• B. $A\supset B$
• C. $A=B$
• D. $A\subset B\text{'}$

Answer 1

The correct option is C

$A=B$

Explanation for the correct answer:

We draw the Venn Diagram for $A=B\cap C$

Now we draw the Venn diagram for $A\cap C$

From the Venn diagram we see that the highlighted part is $A\cap C$

It is given that $B=C\cap A$

From both the given diagram we can conclude that $A=B$

Explanation for Venn Diagrams:

Let $B=\left\{w,x,y,z\right\}$ and $C=\left\{a,b,c,x\right\}$

$A=B\cap C$

$\Rightarrow$ $A=\left\{x\right\}$ $...\left(i\right)$

Now $A\cap C=\left\{x\right\}$

But we know that $B=C\cap A$

$\Rightarrow$ $B=\left\{x\right\}$ $...\left(ii\right)$

From $\left(i\right),\left(ii\right)$ we can deduce that

$\Rightarrow$$A=B$

Hence, option $\left(C\right)$ is the correct answer.