Question

Which of the following is logically equivalent to $~\left(~p\Rightarrow q\right)$?

• A. $p\wedge q$
• B. $p\wedge ~q$
• C. $~p\wedge q$
• D. $~p\wedge ~q$

Answer 1

The correct option is D

$~p\wedge ~q$

Truth table of $p\Rightarrow q$

$p$	$q$	$p\Rightarrow q$
T	T	T
T	F	F
F	T	T
F	F	T

Truth table of $~p$

$p$	$~p$
T	F
F	T

Truth table of $p\wedge q$.

$p$	$q$	$\left(p\wedge q\right)$
T	T	T
T	F	F
F	T	F
F	F	F

Given logical statement is $~\left(~p\Rightarrow q\right)$.

The truth table of the given statement is:

$p$	$q$	$~p$	$\left(~p\Rightarrow q\right)$	$~\left(~p\Rightarrow q\right)$
T	T	F	T	F
T	F	F	T	F
F	T	T	T	F
F	F	T	F	T

The explanation for the correct option:

The truth table of the correct option is:

$p$	$q$	$~p$	$~q$	$~p\wedge ~q$
T	T	F	F	F
T	F	F	T	F
F	T	T	F	F
F	F	T	T	T

The outputs of $~p\wedge ~q$ are the same as the outputs of $~\left(~p\Rightarrow q\right)$.

Hence, the correct option is $D)~p\wedge ~q$.

The explanation for the incorrect options:

The truth table of the incorrect options is:

$p$	$q$	$~p$	$~q$	$p\wedge q$	$p\wedge ~q$	$~p\wedge q$
T	T	F	F	T	F	F
T	F	F	T	F	T	F
F	T	T	F	F	F	T
F	F	T	T	F	F	F

The outputs of the other three options are not the same as the outputs of $~\left(~p\Rightarrow q\right)$.

Hence, the incorrect options are $A),B)$ and $C)$.