Question

The negative of the compound proposition $p\vee \left(~p\vee q\right)$ is

• A. $(p\wedge ~q)\vee ~p$
• B. $~p\wedge \left(p\wedge ~q\right)$
• C. None of these

Answer 1

The correct option is B

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

Explanation for the correct answer:

We are required to find the negation of $p\vee \left(~p\vee q\right)$

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

$=~\left(\left(p\vee ~p\right)\vee \left(p\vee q\right)\right)$ $\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{[}\mathbf{a}\mathbf{\vee }\left(b\wedge c\right)\mathbf{=}\left(a\vee b\right)\mathbf{\wedge }\mathbf{(}\mathbf{a}\mathbf{\vee }\mathbf{c}\mathbf{\left)}\mathbf{\right]}$ …[distributive law]

$=\left(~p\wedge p\right)\wedge \left(~p\wedge ~q\right)$ $\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{[}\mathbf{~}\mathbf{(}\mathbf{a}\mathbf{\wedge }\mathbf{~}\mathbf{b}\mathbf{)}\mathbf{=}\left(~a\vee b\right)\mathbf{]}$ …[complement law]

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

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

Therefore,option (B) is the correct answer.