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

Let us analyze the given assertion $S: A \rightarrow B$, where:

• Antecedent: $A = (P \wedge Q) \rightarrow R$

• Consequent: $B = (P \wedge Q) \rightarrow (Q \rightarrow R)$

Using propositional equivalence rules, we can simplify the consequent $B$:

$$\begin{aligned} B &\equiv (P \wedge Q) \rightarrow (Q \rightarrow R) \\ &\equiv eg(P \wedge Q) \vee ( eg Q \vee R) \\ &\equiv ( eg P \vee eg Q) \vee eg Q \vee R \\ &\equiv eg P \vee eg Q \vee R \\ &\equiv eg(P \wedge Q) \vee R \\ &\equiv (P \wedge Q) \rightarrow R \end{aligned}$$

Thus, the antecedent $A$ and consequent $B$ are logically equivalent ($A \equiv B$). This makes (D) correct.

Now, substituting $B \equiv A$ into the formula $S$:

$$S \equiv A \rightarrow A \equiv eg A \vee A \equiv \text{True}$$

Therefore, $S$ evaluates to true for all truth assignments, which means $S$ is a tautology. This makes (B) correct.