Here are the solutions for the provided questions:

1. Which rule of inference is used in the argument: “If it is Wednesday then the Smart mart will be crowded. It is Wednesday. Thus, the Smart mart is crowded.”

   • Answer: The rule of inference used here is Modus Ponens (option b). It follows the structure: If $p \rightarrow q$, and $p$ is true, then $q$ must be true.

2. Which rule of inference is used in the argument: “If it hails today the local office will be closed. The local office is not closed today. Thus, it did not hail today.”

   • Answer: This follows Modus Tollens (option a), where if $p \rightarrow q$, and $q$ is false, then $p$ must also be false.

3. Which of the following is/are tautology:

   • Answer: None of the given statements are tautologies (option d). A tautology is a statement that is always true regardless of the truth values of its components, and none of these satisfy that.

4. $P \rightarrow (Q \rightarrow R)$ is equivalent to:

   • Answer: $(P \wedge Q) \rightarrow R$ (option A). This is based on the logical equivalence between implications.

5. Proposition $P_1 = (p \vee \neg q) \wedge (q \rightarrow r) \vee (r \vee p)$:

   • Answer: $P_1$ is Contingency (option b), meaning that its truth depends on the truth values of its components.

Would you like a detailed explanation for any of these answers? Here are five related questions:

1. How does Modus Ponens differ from Modus Tollens in terms of logical structure?
2. Can you explain what a tautology is and give an example of one?
3. How do we prove logical equivalences in propositional logic?
4. What is a contingency, and how can it be identified in a logical expression?
5. What are some common logical equivalences involving implication ($\rightarrow$)?

Tip: Logical proofs often use equivalences and rules like Modus Ponens and Modus Tollens. Mastering these can help in solving more complex logical statements.