Let $p$ and $q$ be two propositions. Consider the following two formulae in propositional logic. $$S_1 : \quad ( eg p \land (p \lor q)) \rightarrow q$$ $$S_2 : \quad q \rightarrow ( eg p \land (p \lor q))$$ Which one of the following choices is correct?

First, simplify the antecedent/consequent expression $eg p \land (p \lor q)$ using the distributive law: $$eg p \land (p \lor q) \equiv ( eg p \land p) \lor ( eg p \land q) \equiv \text{False} \lor ( eg p \land q) \equiv eg p \land q$$

Now, evaluate statement $S_1$: $$S_1 : ( eg p \land q) \rightarrow q$$ $$\equiv eg ( eg p \land q) \lor q$$ $$\equiv (p \lor eg q) \lor q$$ $$\equiv p \lor ( eg q \lor q)$$

Therefore, $S_1$ is a tautology.

Next, evaluate statement $S_2$: $$S_2 : q \rightarrow ( eg p \land q)$$ Consider the truth assignment where $p = \text{True}$ and $q = \text{True}$:

• Left-hand side: $q = \text{True}$

• Right-hand side: $

eg p \land q = $\text{False}$ \land $\text{True}$ = $\text{False}$$

• Implication: $\text{True} \rightarrow \text{False} \equiv \text{False}$

Since $S_2$ evaluates to false when both $p$ and $q$ are true, $S_2$ is not a tautology.

Thus, $S_1$ is a tautology but $S_2$ is not a tautology, which corresponds to option (B).