Question

The negation of $p\to (q\wedge r)$

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

Answer 1

The correct option is D

$p\wedge (~q\vee ~r)$

Explanation for the correct option:

Given: $p\to (q\wedge r)$

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 $p\to (q\wedge r)$ is,

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

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

Hence, option D is the correct option.