Question

Which of the following statements is a tautology?

• A. $p \vee( p \wedge q )$
• B. $( p \wedge( p \rightarrow q )) \rightarrow \sim q$
• C. $( p \wedge q ) \rightarrow(\sim( p ) \rightarrow q )$
• D. $p \rightarrow( p \wedge( p \rightarrow q ))$

Answer 1

The Correct Option is C

(i) $p\to (p\Lambda (p\to q))$
$(\sim p)V(p\Lambda (\sim pVq))$
$(\sim p)V(fV(p\Lambda q))$
$\sim pV(p\Lambda q)=(\sim pVp)\Lambda (\sim pVq)$
$=\sim pVq$
(ii) $(p\Lambda q)\to (\sim p\to q)$
$\sim (p\Lambda q)V(pVq)=t$
$\{a,b,d\}V\{a,b,c\}=V$
Tautology
(iii) $(p\Lambda (p\to q))\to \sim q$
$\sim (p\Lambda (\sim pVq))V\sim q=\sim (p\Lambda q)V\sim q=\sim pV\sim q$
Not tantology
(iv) $pV(p\Lambda q)=p$
Not tautology.

So, the correct option is (C): $( p \wedge q ) \rightarrow(\sim( p ) \rightarrow q )$