Question

Consider the statement, “If it is raining, then sky is not filled with clouds.”
Which among the following is TRUE?

• A. Contrapositive of the given statement is, “If sky is filled with clouds, then it is raining.”
• B. Converse of the given statement is, “If sky is filled with clouds, then it is raining.”
• C. Inverse of the given statement is, “If sky is not filled with clouds, then it is raining.”
• D. Negation of the given statement is, “It is raining and sky is filled with clouds.”

Answer 1

The correct option is D Negation of the given statement is, “It is raining and sky is filled with clouds.”

Let $p:$ It is raining.
and $q:$ Sky is not filled with clouds.
The given statement is $p\to q$

As we know that contrapositive of $p\to q$ is $\sim q\to \sim p$
$\Rightarrow$ Contrapositive is “If sky is filled with clouds, then it is not raining.”

Converse of $p\to q$ is $q\to p$
$\Rightarrow$ Converse is “If sky is not filled with clouds, then it is raining.”

Inverse of $p\to q$ is $\sim p\to \sim q$
$\Rightarrow$ Inverse is “If it is not raining, then sky is filled with clouds.”

Negation of $p\to q\equiv (\sim p\vee q)$ is $\sim (\sim p\vee q)=p\wedge \sim q$
$\Rightarrow$ Negation is “It is raining and sky is filled with clouds.”