Question

The number of values of $r\in \{p,q,\sim p,\sim q\}$ for which $((p\wedge q)\Rightarrow (r\vee q))\wedge ((p\wedge r)\Rightarrow q)$ is a tautology, is :

• A. 1
• B. 2
• C. 4
• D. 3

Answer 1

Answer: (B)

$((p\wedge q)\Rightarrow (r\vee q))\wedge ((p\wedge r)\Rightarrow q)$
We know, $p\Rightarrow q$ is equivalent to
$\sim p\vee q$
$(\sim (p\wedge q)\vee (r\vee q))\wedge (\sim (p\wedge r))\vee q))$
$\Rightarrow (\sim p\vee \sim q\vee r\vee q)\wedge (\sim p\vee \sim r\vee q)$
$\Rightarrow (\sim p\vee r\vee t)\wedge (\sim p\vee \sim r\vee q)$
$\Rightarrow (t)\wedge (\sim p\vee \sim r\vee q)$
For this to be tautology, $(\sim p\vee \sim r\vee q)$ must be always true which follows for $r=\sim p$ or $r=q$.