Question

The negation of $p\to (~p\vee q)$ is

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

Answer 1

The correct option is D

$p\wedge ~q$

Explanation for the correct option:

Given: $p\to (~p\vee q)$

We know that the negation of $A$ is given by $~A$ and the De’ Morgan’s laws says $~(a\vee b)=~a\wedge ~b$.

So the negation of $p\to (~p\vee q)$ is,

$$\begin{gathered} ~\left(p\to (~p\vee q)\right)=p\wedge ~(~p\vee q)\left[\because ~(a\to b)=a\wedge ~b\right] \\ =p\wedge (~(~p)\wedge ~q)\left[\because ~(a\vee b)=~a\wedge ~b\right] \\ =p\wedge (p\wedge ~q)\left[\because ~(~a)=a\right] \\ =\left(p\wedge p\right)\wedge ~q \\ =p\wedge ~q\left[\because p\wedge p=p\right] \end{gathered}$$

Therefore, the negation of $p\to (~p\vee q)$ is $p\wedge ~q$.

Hence, option D is the correct option.