Question

Given the following two statements:
$\left(S_1\right):(q\vee p)\to (p\leftrightarrow \sim q)$ is a tautology.
$\left(S_2\right):\sim q\wedge (\sim p\leftrightarrow q)$ is a fallacy. Then

• A. Only $\left(S_1\right)$ is correct
• B. Both $\left(S_1\right)$ and $\left(S_2\right)$ are correct
• C. Only $\left(S_2\right)$ is correct
• D. Both $\left(S_1\right)$ and $\left(S_2\right)$ are not correct

Answer 1

The correct option is D Both $\left(S_1\right)$ and $\left(S_2\right)$ are not correct

The truth table of $S_1$ is given below:
$\begin{array}{cccccc}p& q& \sim q& q\vee p& (p\leftrightarrow \sim q)& (q\vee p)\to (p\leftrightarrow \sim q)\\ T& T& F& T& F& F\\ T& F& T& T& T& T\\ F& T& F& T& T& T\\ F& F& T& F& F& T\end{array}$
$\therefore S_1$ is not correct

The truth table of $S_2$ is given below:
$\begin{array}{cccccc}p& q& \sim q& \sim p& \sim p\leftrightarrow q& \sim q\wedge (\sim p\leftrightarrow q)\\ T& T& F& F& F& F\\ T& F& T& F& T& T\\ F& T& F& T& T& F\\ F& F& T& T& F& F\end{array}$
$\therefore S_2$ is not correct