Question

Let $A,B$ and $C$ be sets such that $\varphi \ne A\cap B\subseteq C$. Then which of the following statements is not true ?

• A. $B\cap C\ne \varphi$
• B. $(C\cup A)\cap (C\cup B)=C$
• C. If $(A-B)\subseteq C,$ then $A\subseteq C$
• D. If $(A-C)\subseteq B,$ then $A\subseteq B$

Answer 1

The correct option is D If $(A-C)\subseteq B,$ then $A\subseteq B$

Let $A=\{1,2,3\}B=\{2,3,4\}C=\{1,2,3,5,6\}$
Here, $A\cap B=\{2,3\}\subseteq C$
we can clearly see that
$A-C=\varphi \subseteq B$
But $A⊈B$ So, option $\left(d\right)$ is false

$(A-B)=\left\{1\right\}\subseteq C,$ and $\because A\cap B=\{2,3\}\subseteq C$ , therefore $A\subseteq C$

$(C\cup A)\cap (C\cup B)=\{1,2,3,5,6\}\cap \{1,2,3,4,5,6\}=C$

$B\cap C=\{2,3\}\ne \varphi$