Question

The contrapositive of the statement ‘ If it is a Friday or if the sky is clear, then Sam plays’ is ___.

• A. If Sam plays, then it is a friday or the sky is clear
• B. If Sam does not play, then it is a Friday or the sky is clear
• C. If Sam does not play, then it is not a Friday and the sky is not clear
• D. If Sam does not play, then it is a Friday or the sky is not clear

Answer 1

The correct option is C If Sam does not play, then it is not a Friday and the sky is not clear

The contrapositive of $p\Rightarrow q$ is $\sim q\Rightarrow \sim p.$
p: It is a Friday
q: The sky is clear
r: Sam plays
The given statement is $(p\vee q)\Rightarrow r$. Its contrapositive is $\sim r\Rightarrow \sim (p\vee q)$. Also, $\sim (p\vee q)=\sim p\wedge \sim q$. Hence, the contrapositive of the given statement is $\sim r\Rightarrow \sim p\wedge \sim q$
This can be translated as ‘If Sam does not play, then it is not a Friday and the sky is not clear’.