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 $r=q$.
