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

Step 1: Analyse the structure of the implication.
The given statement is of the form $A \rightarrow B$, where \[ A = (P \land Q) \rightarrow R \text{and} B = (P \land Q) \rightarrow (Q \rightarrow R). \]

Step 2: Simplify the consequent $B$.
Recall that $Q \rightarrow R \equiv \neg Q \lor R$. Hence, \[ (P \land Q) \rightarrow (Q \rightarrow R) \equiv (P \land Q) \rightarrow (\neg Q \lor R). \] Whenever $(P \land Q)$ is true, $Q$ is true, so $(\neg Q \lor R)$ reduces to $R$. Thus, \[ (P \land Q) \rightarrow (Q \rightarrow R) \equiv (P \land Q) \rightarrow R. \]

Step 3: Establish equivalence of antecedent and consequent.
From the above step, we see that \[ A \equiv B. \] Hence, the antecedent of $S$ is logically equivalent to the consequent of $S$.

Step 4: Determine the nature of $S$.
Since $S$ has the form $A \rightarrow A$, it is always true regardless of the truth values of $P$, $Q$, and $R$. Therefore, $S$ is a tautology.

Step 5: Final conclusion.
Thus, $S$ is a tautology, and its antecedent is logically equivalent to its consequent.