Question

Let $A,B,C$ and $D$ be four non-empty sets. The contrapositive statement of "If $A\subseteq B$ and $B\subseteq D$, then $A\subseteq C$" is :

• A. If $A\subseteq C$, then $B\subset A$ or $D\subset B$
• B. If $A⊈C$, then $A\subseteq B$ and $B\subseteq D$
• C. If $A⊈C$, then $A⊈B$ and $B\subseteq D$
• D. If $A⊈C$, then $A⊈B$ or $B⊈D$

Answer 1

The correct option is D If $A⊈C$, then $A⊈B$ or $B⊈D$

Let the statements $A\subseteq B$ be $p$,
$B\subseteq D$ be $q$
and $A\subseteq C$ be $r$.
The logical statement is $(p\wedge q)\Rightarrow r$
The contrapositive of above statement is $\sim r\Rightarrow \sim (p\wedge q)$
i.e, $\sim r\Rightarrow (\sim p\vee \sim q)$
$\therefore$ The contrapositive of the given statement is ''If $A⊈C$, then $A⊈B$ or $B⊈D$