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Question

Mathematics Question on mathematical reasoning

Which of the following statements is a tautology?

A

p∨(p∧q)p \vee( p \wedge q )

B

(p∧(p→q))→∼q( p \wedge( p \rightarrow q )) \rightarrow \sim q

C

(p∧q)→(∼(p)→q)( p \wedge q ) \rightarrow(\sim( p ) \rightarrow q )

D

p→(p∧(p→q))p \rightarrow( p \wedge( p \rightarrow q ))

Answer

(p∧q)→(∼(p)→q)( p \wedge q ) \rightarrow(\sim( p ) \rightarrow q )

Explanation

Solution

(i) p→(pΛ(p→q))
(∼p)V(pΛ(∼pVq))
(∼p)V(fV(pΛq))
∼pV(pΛq)=(∼pVp)Λ(∼pVq)
=∼pVq
(ii) (pΛq)→(∼p→q)
∼(pΛq)V(pVq)=t
{a,b,d}V{a,b,c}=V
Tautology
(iii) (pΛ(p→q))→∼q
∼(pΛ(∼pVq))V∼q=∼(pΛq)V∼q=∼pV∼q
Not tantology
(iv) pV(pΛq)=p
Not tautology.

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