Question

Which of the following is a contradiction?

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

Answer 1

The correct option is A

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

Explanation for the correct option:

The possible truth table is,

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

All outputs of $\left(p\wedge q\right)\wedge ~\left(p\vee q\right)$ are false.

So, it is a contradiction.

Hence, the correct option is $A)\left(p\wedge q\right)\wedge ~\left(p\vee q\right)$.