Mathematical Reasoning Question 10

Question 10 - 31 January - Shift 2

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:

(1) 3

(2) 2

(3) 1

(4) 4

Show Answer

Answer: (2)

Solution:

$((p \wedge q) \Rightarrow(r \vee q)) \wedge((p \wedge r) \Rightarrow q)$

We know, $p \Rightarrow q$ is equivalent to

$\sim p \vee q$

$(\sim(p \wedge q) v(r \vee q)) \wedge(\sim(p \wedge r)) \vee q))$

$\Rightarrow(\sim p \vee \sim q \vee r \vee q) \wedge(\sim p \vee \sim r \vee q)$

$\Rightarrow(\sim p \vee r \vee t) \wedge(\sim p \vee \sim r \vee q)$

$\Rightarrow(t) \wedge(\sim p \vee \sim r \vee q)$

For this to be tautology, $(\sim p \vee \sim r \vee q)$ must be always true which follows for $r=\sim p$ or $r=q$.