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 $r = q$.