QJEEM19Ma729b: JEE Main 2019 Mathematics - Class 11 JEEM19Ma729b Solution: Which one of the following statements is not a tautology?

Here you have to check all the options.

We know, p $\to$ q = $\sim$ p $\vee$ q

(A) (p $\wedge$ q) $\to$ (~ p) $\vee$ q

= $\sim$ (p $\wedge$ q) $\vee$ ((~ p) $\vee$ q)

= ( $\sim$ p $\vee$ $\sim$ q) $\vee$ ((~ p) $\vee$ q)

= $\sim$ p $\vee$ (q $\vee$ $\sim$ q)

= $\sim$ p $\vee$ t

[(q $\vee$ $\sim$ q) = t (tautology)]

= t

(B) (p $\wedge$ q) $\to$ p

= $\sim$ (p $\wedge$ q) $\vee$ p

= $\sim$ p $\vee$ $\sim$ q $\vee$ p

= t $\vee$ $\sim$ q

= t

(C) ( p $\vee$ q) $\to$ ( p $\vee$ (~q))

= $\sim$ ( p $\vee$ q) $\vee$ ( p $\vee$ (~q))

= ( $\sim$ p $\wedge$ $\sim$ q) $\vee$ ( p $\vee$ (~q))

= $\sim$ q $\wedge$ (p $\vee$ $\sim$ p)

= $\sim$ q $\wedge$ t

= $\sim$ q (not tautology)

(D) p $\to$ ( p $\vee$ q)

= $\sim$ p $\vee$ ( p $\vee$ q)

= t $\vee$ q

= t