Question

For the statements $p$ and $q$, consider the following compound statements:

(a) $(~q\wedge (p\to q\left)\right)\to ~p$

(b) $\left(\right(p\vee q\left)\right)\wedge ~p)\to q$

Then which of the following statements 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

Explanation for the correct option:

Step 1. Consider statement (a):

$p$	$q$	$~q$	$p\to q$	$~q\wedge \left(p\to q\right)$	$~p$	$\left(~q\wedge \left(p\to q\right)\right)\to ~p$
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

(a) is tautology.

Step 2. Consider statement (b):

$p$	$q$	$p\vee q$	$~p$	$\left(p\vee q\right)\wedge ~p$	$\left(\left(p\vee q\right)\wedge ~p\right)\to q$
T	T	T	F	F	T
T	F	T	F	F	T
F	T	T	T	T	T
F	F	F	T	F	T

(b) is tautology.

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

Hence, Option ‘C’ is Correct.