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Q. Which of the following statements is a tautology?

JEE MainJEE Main 2023Mathematical Reasoning

Solution:

(i) $p \rightarrow( p \Lambda( p \rightarrow q ))$
$(\sim p ) V ( p \Lambda(\sim p V q))$
$(\sim p) V(f V(p \Lambda q))$
$\sim p V ( p \Lambda q )=(\sim p V p ) \Lambda(\sim p Vq )$
$=\sim p V q$
(ii) $( p \Lambda q ) \rightarrow(\sim p \rightarrow q )$
$\sim( p \Lambda q ) V ( p V q )= t$
$\{ a , b , d \} V \{ a , b , c \}= V$
Tautology
(iii) $( p \Lambda( p \rightarrow q )) \rightarrow \sim q$
$\sim( p \Lambda(\sim p V q )) V \sim q =\sim( p \Lambda q ) V \sim q =\sim p V \sim q$
Not tantology
(iv) $p V ( p \Lambda q )= p$
Not tautology.