Question

Is [(~p → q) ∧ (r → ~p)] a tautology?

Option 1: Yes, it is a tautology.
Option 2: No, it is not a tautology.
Option 3: It depends on the values of p, q, and r.
Option 4: Not enough information to determine.

Answer 1

After analyzing the given logical statement, we can conclude that it is not a tautology because its truth depends on the values of p, q, and r. Consequently, it can be false in certain cases and therefore it does not hold true in all possible interpretations.

Step-by-step explanation:

The statement in question is: (¬p → q) ∧ (r → ¬p). To determine if it is a tautology, we need to analyze the truth table for all possible truth values of p, q, and r.

Let's break down the statement:

• ¬p → q means 'if not p, then q'.
• r → ¬p means 'if r, then not p'.
• The whole statement is the conjunction (AND) of the two.

To be a tautology, the statement must be true for all possible truth values of p, q, and r. After constructing the truth table, we can determine if there's any row in which the whole statement is false. If we find even one row where the statement is false, then it is not a tautology.

Analyzing the logical form of the argument suggests it depends on the truth values of p, q, and r. Therefore, the correct answer is:

Option 2: No, it is not a tautology.

A tautology is a statement that is true under any interpretation or assignment of truth values to its components. Since the truth of the given compound statement depends on the values of p, q, and r, it can be false in some cases, which means it is not a tautology.