Question

Write the negation of the statement "If the switch is on, then the fan rotates".\[\]
A. “If the switch is not on then the fan does not rotate”\[\]
B. “If the fan does not rotate, then the switch is not on.”\[\]
C. “The switch is not on or the fan rotates.” \[\]
D. “The switch is on and the fan does not rotate.” \[\]

Answer 1

\[\] Hint: We see that the given statement is in the form if $p$ then $q$ or $p\to q$.We denote the antecedent in the given statement “the switch is on” as $p$ and “the fan rotates” as$q$. The negation of $p\to q$ is equivalent to $p\wedge \tilde{\ }q$. We check option for the correct connectives.\[\]

Complete step by step answer:
We know from the mathematical logic that if the statement $p$ has a truth value T or F then the negation of $p$ is denoted as $\tilde{\ }p$ and has truth value F or T respectively. The through table of $\tilde{\ }p$ is given as
\[\begin{matrix}
   p & \tilde{\ }p \\
   T & F \\
   F & T \\
\end{matrix}\]

We also know that statement with implication (with logical connective If...then...) of their truth values is denoted as $p\to q$ and has a truth value F only when one of $p$ has a truth value T and $q$ has a truth value $F$ otherwise true. It is also expressed as If $p$ then $q$.The statement $p$ is called the antecedent and $q$is called consequence.
The truth table of $p\to q$ is
\[\begin{matrix}
   p & q & p\to q \\
   T & T & T \\
   T & F & F \\
   F & T & T \\
   F & F & T \\
\end{matrix}\]
The negation of the statement $p\to q$ means antecedent $p$ is true and consequence $q$ is false. We can write it as $\tilde{\ }\left( p\to q \right)\equiv p\wedge \tilde{\ }q$.
\[\begin{matrix}
   p & q & \tilde{\ }q & p\to q & \tilde{\ }\left( p\to q \right) & p\wedge \tilde{\ }q \\
   T & T & F & T & F & F \\
   T & F & T & F & T & T \\
   F & T & F & T & F & F \\
   F & F & T & T & F & F \\
\end{matrix}\]
The given statement is “If the switch is on, then the fan rotates.” We see that the statement is in an implicit form where we denote the stamen “the switch is on” as $p$ and “the fan rotates” as $q$. Then we can have $\tilde{\ }p$ as “the switch is not on” and $\tilde{\ }q$ as “the fan does not rotate”. The negation of the statement $p\to q$ is $p\wedge \tilde{\ }q$ which we can write as “The switch is on and the fan does not rotate.”\[\]
We look at the options and find the same statement at option D.\[\]

Note:
We take note not to confuse the negation of $p\to q$ with the inverse of the statement $p\to q$ which is given by $\tilde{\ }p\to \tilde{\ }q$, the converse of the statement $q\to p$ or the contrapositive of the statement $\tilde{\ }q\to \tilde{\ }p$.