Question

Which of the following is a contradiction?
A) $p \vee q$
B) $p \wedge q$
C) $p \vee (\neg p)$
D) $p \wedge (\neg p)$

Answer 1

We can use truth tables to find the answer. A compound statement is a contradiction, if it is always false. So we can check the truth tables in each case.

Complete step-by-step answer:
Given two statements $p$ and $q$. In the options we have their different combinations. We have to find which of the compound statements given is a contradiction.
A compound statement is a contradiction, if it is always false.
Consider the truth table.
For two statements $p$ and $q$, we have four cases.
Both are true, both are false and one is true and the other is false.
We can represent a true statement by $T$ and a false statement by $F$.
We know $p \vee q$ is true if either one is true and is false if both are false.
Also $p \wedge q$ is true only if both are true and false if either one is false.
So we have,

$p$	$q$	$p \vee q$	$p \wedge q$
$T$	$T$	$T$	$T$
$T$	$F$	$T$	$F$
$F$	$T$	$T$	$F$
$F$	$F$	$F$	$F$

From the truth table, we can infer that $p \vee q$ and $p \wedge q$ are not contradictions, since all the cases are not false here.
So let us check $p \vee (\neg p)$ and $p \wedge (\neg p)$.
Consider the truth table in this case.

$p$	$\neg p$	$p \vee (\neg p)$	$p \wedge (\neg p)$
$T$	$F$	$T$	$F$
$F$	$T$	$T$	$F$

Here we can see $p \vee \neg p$ is not a contradiction.
But $p \wedge \neg p$ is a contradiction since it is false in all cases.

$\therefore$ The answer is option D.

Note: We can also find the answer without a truth table. We can simply infer by logic that a statement and its inverse cannot be true at the same time. So clearly $p \wedge \neg p$ is a contradiction.