Question

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

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

Answer 1

The correct option is C

$\left(~p\wedge q\right)\vee ~q$

Explanation for the correct option:

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

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 $(p\vee ~q)\wedge q$ is,

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

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

Hence, option C is the correct option.