Question

For the statements $p$ and $q,$ consider the following compound statements :
(a) $(\sim q\wedge (p\to q\left)\right)\to \sim p$
(b) $\left(\right(p\vee q)\wedge \sim p)\to q$
Then which of the following ststements is correct ?

• A. (a) is a tautology but not (b).
• B. (a) and (b) both are not tautologies.
• C. (a) and (b) both are tautologies.
• D. (b) is a tautology but not (a).

Answer 1

The correct option is C (a) and (b) both are tautologies.

(a) $\begin{array}{ccccccc}p& q& \sim q& p\to q& \sim q\wedge (p\to q)& \sim p& (\sim q\wedge (p\to q\left)\right)\to \sim p\\ T& T& F& T& F& F& T\\ T& F& T& F& F& F& T\\ F& T& F& T& F& T& T\\ F& F& T& T& T& T& T\end{array}$

(a) is tautology.

(b) $\begin{array}{cccccc}p& q& p\vee q& \sim p& (p\vee q)\wedge \sim p& \left(\right(p\vee q)\wedge \sim p)\to q\\ T& T& T& F& F& T\\ T& F& T& F& F& T\\ F& T& T& T& T& T\\ F& F& F& T& F& T\end{array}$

(b) is tautology.
$\therefore$ (a) and (b) both are tautologies.