Question

The statement $(p \land (p \to q) \land (q \to r)) \to r$ is :

• A. a tautology
• B. a fallacy
• C. equivalent to $p \to \sim r$
• D. equivalent to $q \to \sim r$

Hint:

Use the Modus Ponens rule: $[p \land (p \to q)] \implies q$. Applying it repeatedly here, $[p \land (p \to q)]$ gives $q$, and $[q \land (q \to r)]$ gives $r$. Thus, the antecedent implies the consequent $r$, confirming it's a tautology.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In mathematical logic, we can simplify statements using logical equivalence laws or truth tables. A tautology is a statement that is true for all possible truth values of its components.
Step 2: Detailed Explanation:
Let's simplify the antecedent: $(p \land (p \to q) \land (q \to r))$.
1. $p \land (p \to q) \equiv p \land (\sim p \lor q) \equiv (p \land \sim p) \lor (p \land q) \equiv F \lor (p \land q) \equiv p \land q$.
2. Now, $(p \land q) \land (q \to r) \equiv p \land (q \land (\sim q \lor r)) \equiv p \land ((q \land \sim q) \lor (q \land r)) \equiv p \land (F \lor (q \land r)) \equiv p \land q \land r$.
The whole statement is $(p \land q \land r) \to r$.
Using the conditional law $X \to Y \equiv \sim X \lor Y$:
\[ \sim (p \land q \land r) \lor r \equiv (\sim p \lor \sim q \lor \sim r) \lor r \]
\[ \equiv \sim p \lor \sim q \lor (\sim r \lor r) \equiv \sim p \lor \sim q \lor T \equiv T \]
Since the final result is always True ($T$), the statement is a tautology.
Step 3: Final Answer:
The statement is a tautology.