Question

Choose the correct choice(s) regarding the following propositional logic assertion $S$: \[ S : \big((P \land Q) \rightarrow R\big) \rightarrow \big((P \land Q) \rightarrow (Q \rightarrow R)\big) \]

• A. $S$ is neither a tautology nor a contradiction.
• B. $S$ is a tautology.
• C. $S$ is a contradiction.
• D. The antecedent of $S$ is logically equivalent to the consequent of $S$.

Hint:

Any propositional formula of the form $A \rightarrow A$ is always a tautology.

Answer 1

The Correct Option is B, D

To analyze the given propositional logic assertion \(S\), we must determine whether it is a tautology, contradiction, or neither. We are also asked to verify if the antecedent of \(S\) is logically equivalent to its consequent. Here is a step-by-step breakdown:

The proposition is given by: \[ S : \big((P \land Q) \rightarrow R\big) \rightarrow \big((P \land Q) \rightarrow (Q \rightarrow R)\big) \]

1. The proposition \(S\) is of the form \(A \rightarrow B\) where:
   • \(A = (P \land Q) \rightarrow R\) 
   • \(B = (P \land Q) \rightarrow (Q \rightarrow R)\)
2. A statement of form \(X \rightarrow Y\) in logic is considered a tautology if it always evaluates to true, regardless of the truth values of \(X\) and \(Y\).
3. Let's examine when \(A\) and \(B\) are true:
   • \(A = (P \land Q) \rightarrow R\) is true when:
     • Either \((P \land Q)\) is false, or
     • \(R\) is true.
   • \(B = (P \land Q) \rightarrow (Q \rightarrow R)\) is true when:
     • Either \((P \land Q)\) is false, or
     • \(Q \rightarrow R\) is true, i.e., \(Q\) is false or \(R\) is true.
4. Notice that both \(A\) and \(B\) have similar conditions for being true. Specifically, if \((P \land Q)\) is true, then for both \(A\) and \(B\) to be true, \(R\) must be true.
5. Thus, in all scenarios, when \(A\) is true, \(B\) is also true. This implies that \(A\) logically implies \(B\) in all cases.
6. Therefore, \(S\) is indeed a tautology, because it is true for all possible truth values of \(P\), \(Q\), and \(R\).
7. Furthermore, the logical equivalence of the antecedent and the consequent also holds true, confirming the second correct choice.

Conclusively, both the assertions given in the correct answer:

• \(S\) is a tautology.\)
• \(The antecedent of S is logically equivalent to the consequent of S.\)

are accurate.