(𝑝 ∧ 𝑞) → 𝑟
(𝑟 ∨ 𝑠) → 𝑝
𝑟 ∧ ~𝑝
∴ 𝑠 ∨ 𝑞
You should begin your answer by stating “Valid” or “Invalid”, followed by your proof. You are not
allowed to show your truth table (partial or full) in your answer, though you may use truth table in your
own rough work to help you derive your proof.

**Invalid.**

Here's the proof for why this argument is invalid:

### Premises:
1. \( (p \land q) \to r \)  (Premise 1)
2. \( (r \lor s) \to p \)  (Premise 2)
3. \( r \land \neg p \)    (Premise 3)

### Conclusion to Prove: 
\( s \lor q \)

### Analysis:

#### Step 1: Analyzing Premise 3
From \( r \land \neg p \), we know that:
- \( r \) is true.
- \( p \) is false.

#### Step 2: Implications from Premise 2
Consider \( (r \lor s) \to p \). Since \( p \) is false (from Step 1), the only way for \( (r \lor s) \to p \) to hold true is if \( r \lor s \) is false. Therefore:
- \( r \lor s \) must be false.
- Since \( r \) is true (from Step 1), it follows that \( s \) must be false to make \( r \lor s \) false.

Thus, \( s \) is false.

#### Step 3: Implications from Premise 1
Now consider \( (p \land q) \to r \). Since \( r \) is true (from Step 1), this tells us nothing about \( p \land q \), because the implication is satisfied when the consequent \( r \) is true, regardless of whether \( p \land q \) is true or false.

#### Step 4: Final Conclusion
Given that \( s \) is false (from Step 2), for \( s \lor q \) to be true, \( q \) must be true. However, none of the premises give us any information about the truth value of \( q \). It could be either true or false.

Therefore, the argument does not guarantee the truth of \( s \lor q \). Since the conclusion \( s \lor q \) cannot be definitively derived from the premises, the argument is **invalid**.

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Would you like further details or clarifications? Here are some related questions:

1. How would the truth table for this argument look?
2. Can you prove the argument is valid using natural deduction?
3. What happens if \( p \) is true in Premise 3?
4. Can we modify the premises to make the argument valid?
5. How can we represent this argument symbolically in terms of a proof system?

**Tip:** Always verify the truth of each proposition step by step to see if the conclusion follows logically from the premises.

(𝑝 ∧ 𝑞) → 𝑟
(𝑟 ∨ 𝑠) → 𝑝
𝑟 ∧ ~𝑝
∴ 𝑠 ∨ 𝑞

This problem tests your understanding of propositional logic, specifically working with logical implications, disjunctions, conjunctions, and negations. Explore a detailed proof determining whether the argument is valid or invalid. Suitable for college-level students learning about logical reasoning.

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