Question

$~(p\vee q)\vee (~p\wedge q)$is logically equivalent to

• A. $~p$
• B. $p$
• C. $q$
• D. $~q$

Answer 1

The correct option is A

$~p$

Explanation for the correct option:

Finding $\mathbf{~}\mathbf{}\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{\vee }\mathbf{(}\mathbf{~}\mathbf{}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{}$is logically equivalent to:

$$\begin{gathered} ~\left(p\vee q\right)\vee (~p\wedge q) \\ (~p\wedge ~q)\vee (~p\wedge q) \\ ~p\wedge (~q\vee q)\left[\because p\wedge (q\vee r)=(p\wedge q)\vee (p\wedge r)\right] \\ \therefore ~p\wedge (~q\vee q)\cong ~p\left[\text{If}q\text{is}T~q\text{is}F\Rightarrow ~q\vee q\cong T\right] \end{gathered}$$

Hence the correct answer is option (A)