Question

The statement $\left( {p \wedge q} \right) \wedge \left( {\neg p \vee \neg q} \right)$ is___
A.A tautology
B.A contradiction
C.A contingency
D.Neither a tautology nor a contradiction

Answer 1

Hint: First, we will write the truth table of $\left( {p \wedge q} \right)$, that is the statement is true when both $p$ and $q$ are true, otherwise it is false. Next, write the truth table of $\left( {\neg p \vee \neg q} \right)$. For $\neg p$ the result is opposite. Also, $\left( {p \vee q} \right)$ is false only when \[p\] and $q$ is false. Then, determine the type of statement from the result.

Complete step-by-step answer:
We will first construct a truth table corresponding to the given statement $\left( {p \wedge q} \right) \wedge \left( {~p \vee ~q} \right)$
We will find the truth table of $\left( {p \wedge q} \right)$
We know that $\left( {p \wedge q} \right)$ is true when both $p$ and $q$ are true, otherwise it is false.

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

Now, write the truth table of $\left( {\neg p \vee \neg q} \right)$
$\left( {p \vee q} \right)$ is false only when \[p\] and $q$ is false.

$p$	$q$	$\neg p$	$\neg q$	$\left( {~p \vee ~q} \right)$
T	T	F	F	F
T	F	F	T	T
F	T	T	F	T
F	F	T	T	T

Next, we will write the truth table of $\left( {p \wedge q} \right) \wedge \left( {\neg p \vee \neg q} \right)$

$\left( {p \wedge q} \right)$	$\left( {\neg p \vee \neg q} \right)$	$\left( {p \wedge q} \right) \wedge \left( {~p \vee ~q} \right)$
T	F	F
F	T	F
F	T	F
F	T	F

Since, we have false in every row, then the statement is a contradiction.
Hence, option B is correct.

Note: If the result in every row is true, then the statement is a tautology. If the result is false for every row, then the statement is contradictory. If the proposition is neither a tautology nor a contradiction, then the statement is a contingency.