Question

Which of the following statements is a tautology?

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

Answer 1

Answer: (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.