Question

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

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

Answer 1

Answer: (A)

For $A=C,A-C=\Phi$
$\Rightarrow \Phi \subseteq B$
But $A⊈B$
$\Rightarrow$ option 1 is NOT true
$Letx\in (Cx\in (C\cup A)\cap (C\cup B))$
$\Rightarrow x\in (C\cup A)andx\in (C\cup B)$
$\Rightarrow (x\in Corx\in A)and(x\in Corx\in B)$
$\Rightarrow x\in Corx\in (A\cap B)$
$$\begin{gathered} \Rightarrow x\in Corx\in C(asA\cup B \\ subseteqC \end{gathered}$$
$\Rightarrow x\in C$
$\Rightarrow (C\cup A)\cap (C\cup B)\subseteq C(1)$
$Nowx\in C\Rightarrow x\in (C\cup A)andx\in (C\cup B)$
$\Rightarrow x\in (C\cup A)\cap (C\cup B)$
$\Rightarrow C\subseteq (C\cup A)\cap (C\cup B)(2)$
$\Rightarrow from(1)and(2)$
$C=(C\cup A)\cap (C\cup B)$
$\Rightarrow$ option 2 is true
Let x $\in$A and $x\notin B$
$\Rightarrow x\in (A-B)$
$\Rightarrow x\in C(asA-B\subseteq C)$
Let x $\in Aandx\in B$
$\Rightarrow x\in (A\cap B)$
$\Rightarrow x\in C(asA\cap B\subseteq C$
Hence x $\in$ A $\Rightarrow$ x $\in$ C
$\Rightarrow A\subseteq C$
$\Rightarrow$ Option 3 is true
as $C\supseteq (A\cap B)$
$\Rightarrow B\cap C\supseteq (A\cap B)$
as $A\cap B\le \Phi$
$\Rightarrow B\cap C\le \Phi$
$\Rightarrow$ Option 4 is true