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Which statement given below is tautology ?
$p \rightarrow( p \Lambda( p \rightarrow q ))$
$( p \Lambda q ) \rightarrow(\sim( p ) \rightarrow q ))$
$( p \Lambda( p \rightarrow q )) \rightarrow \sim q$
$p V ( p \Lambda q )$
Solution
$\text { (i) } p \rightarrow( p \Lambda( p \rightarrow q ))$
$(\sim p ) V ( p \Lambda(\sim p V q ))$
$(\sim p ) V ( f V ( p \Lambda q ))$
$\sim p V ( p \Lambda q )=(\sim p V p ) \Lambda(\sim p V q)$
$=\sim p V q$
$(ii)$ $( p \Lambda q ) \rightarrow(\sim p \rightarrow q )$
$\sim( p \Lambda q ) V ( p V q )= t$
$\{a, b, d\} V\{a, b, c\}=V$
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.
