Question

The proposition \((p \rightarrow \neg p) \wedge (\neg p \rightarrow p)\) is

• A. a tautology
• B. a contradiction
• C. neither tautology nor contradiction
• D. both tautology and contradiction

Hint:

\(p \rightarrow q\) is logically equivalent to \(\neg p \vee q\).

Answer 1

The Correct Option is B

The given proposition is \((p \rightarrow \neg p) \wedge (\neg p \rightarrow p)\). We need to determine if it is a tautology, contradiction, or neither.

Let's start by understanding the basic logical operations involved: 

• The implication \( p \rightarrow q \) is logically equivalent to \( \neg p \vee q \).

Now let's analyze each part of the proposition separately:

Part 1: \( p \rightarrow \neg p \)

• Replacing with equivalence logic: \( \neg p \vee \neg p \equiv \neg p \).

Part 2: \( \neg p \rightarrow p \)

• Replacing with equivalence logic: \( p \vee p \equiv p \).

The given proposition combines both parts using a logical AND:

• \((p \rightarrow \neg p) \wedge (\neg p \rightarrow p) \equiv (\neg p) \wedge (p)\)

This reduces to a conjunction of two opposite literals, \( \neg p \) and \( p \). This can never be true simultaneously, hence:

• If \( p \) is true, then \( \neg p \) is false, making the entire expression false.
• If \( \neg p \) is true, then \( p \) is false, again making the entire expression false.

Therefore, the expression \((\neg p) \wedge (p)\) is a contradiction as it is always false.

Hence, the correct answer is: a contradiction