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Mathematical Reasoning
hard
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
$3$
B
$2$
C
$1$
D
$4$
(JEE MAIN-2023)
Solution
$(( p \wedge q ) \Rightarrow( r \vee q )) \wedge(( p \wedge r ) \Rightarrow q )$
We know, $p \Rightarrow q$ is equivalent to
$-p \vee q$
$(-(p \wedge q) \vee(r \vee q)) \wedge(-(p \wedge r)) \vee q))$
$\Rightarrow(-p \vee-q \vee r \vee q) \wedge(-p \vee-r \vee q)$
$\Rightarrow(-p \vee r \vee t) \wedge(-p \vee-r \vee q)$
$\Rightarrow(t) \wedge(-p \vee-r \vee q)$
For this to be tautology, ( $-p \vee-r \vee q)$ must be always true which follows for $r=-p$ or $r=q$.
Standard 11
Mathematics
