Question

If $p,q,r$ are any three logical statements, then which one of the following is correct?

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

Answer 1

The correct option is C

$~[p\vee (~q\left)\right]\equiv (~p)\wedge q$

Explanation for the correct option

Check for option (C) which is $~\left[p\vee \left(~q\right)\right]\equiv \left(~p\right)\wedge q$

First, solve for LHS and check if it's equal to RHS.

$\begin{array}{rcc}~\left[p\vee \left(~q\right)\right]& =& \left(~p\right)\wedge ~\left(~q\right)\left(\text{using Demorgans Law}\right)\\ & & =~p\wedge q\left(\text{using Double Negation Law}\right)\end{array}$

From this one can conclude that,

$\mathrm{L}.\mathrm{H}.\mathrm{S}=\mathrm{R}.\mathrm{H}.\mathrm{S}$

Therefore option (C) is correct.

Explanation for incorrect options

Check for option (A) which is$~[p\wedge (~q\left)\right]\equiv (~p)\wedge q$

Solve for LHS and check if it's equal to RHS.

$\begin{array}{rcl}~\left[p\wedge \left(~q\right)\right]& =& \left(~p\right)\vee ~\left(~q\right)\left(\text{using Demorgans Law}\right)\\ & & =\left(~p\right)\vee q\left(\text{using Double Negetarion Law}\right)\end{array}$

Here, $\mathrm{L}.\mathrm{H}.\mathrm{S}\ne \mathrm{R}.\mathrm{H}.\mathrm{S}$

Therefore, option (A) is incorrect.

Check for option (B) which is$~(p\vee q)\wedge (~r)\equiv (~p)\vee (~q)\vee (~r)$

Solve for LHS and check if it's equal to RHS.

$\begin{array}{rcl}~(p\vee q)\wedge (~r)& =& \left[(~p)\wedge (~q)\right]\wedge (~r)\left(\text{using Demorgans Law}\right)\\ & =& (~p)\wedge (~q)\wedge (~r)\left(\text{using Double Negetarion Law}\right)\end{array}$

Here as well, $\mathrm{L}.\mathrm{H}.\mathrm{S}\ne \mathrm{R}.\mathrm{H}.\mathrm{S}$

Therefore, option (B) is incorrect.

Check for option (D) which is$~[p\wedge (~q\left)\right]\equiv (~p)\wedge ~q$

Solve for LHS and check if it's equal to RHS.

$\begin{array}{rcl}~[p\wedge (~q\left)\right]& =& (~p)\vee ~(~q)\left(\text{using Demorgans Law}\right)\\ & =& (~p)\vee q\left(\text{using Double Negetarion Law}\right)\end{array}$

We can see that $\mathrm{L}.\mathrm{H}.\mathrm{S}\ne \mathrm{R}.\mathrm{H}.\mathrm{S}$

Therefore, option (D) is incorrect.

Hence option (C) is the correct option.