Prove that R→S is a valid conclusion from the premises ~PVQ, ~QVR, and R→S.

We are tasked with proving that \( R \rightarrow S \) is a valid conclusion from the premises:

1. \( \neg P \vee Q \)  (Premise 1)
2. \( \neg Q \vee R \)  (Premise 2)
3. \( R \rightarrow S \) (Premise 3)

### Step-by-Step Proof

Let's break down the logic step by step.

#### 1. **Premise: \( \neg P \vee Q \)**
This is equivalent to saying, "If \( P \) is false, then \( Q \) must be true."

#### 2. **Premise: \( \neg Q \vee R \)**
This means, "If \( Q \) is false, then \( R \) must be true."

#### 3. **Premise: \( R \rightarrow S \)**
This means, "If \( R \) is true, then \( S \) must be true."

We need to show that \( R \rightarrow S \) is a valid conclusion from the first two premises. The conclusion essentially states that if \( R \) is true, then \( S \) will follow.

### Step-by-Step Proof by Contradiction

We can assume \( R \) is true and show that \( S \) must follow using the premises.

#### Assume \( R \) is true:
- From Premise 3, if \( R \) is true, then \( S \) must be true (direct conclusion from \( R \rightarrow S \)).

#### If \( R \) is false:
We want to show what happens if \( R \) were false. 

1. From Premise 2: \( \neg Q \vee R \), if \( R \) is false, then \( \neg Q \) must be true, meaning \( Q \) is false.
2. If \( Q \) is false, we can go back to Premise 1: \( \neg P \vee Q \). Since \( Q \) is false, \( \neg P \) must be true, meaning \( P \) is false.

At this point, we have \( P \) false, \( Q \) false, and \( R \) false, which contradicts the premise that \( R \rightarrow S \).

#### Therefore, \( R \) must be true.

Once we have shown that \( R \) is true, \( R \rightarrow S \) directly leads to the conclusion that \( S \) is true, which proves that \( R \rightarrow S \) is a valid conclusion.

### Conclusion

Thus, we have shown that \( R \rightarrow S \) logically follows from the premises \( \neg P \vee Q \), \( \neg Q \vee R \), and \( R \rightarrow S \). Therefore, \( R \rightarrow S \) is a valid conclusion.

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Would you like any further clarification on any part of the proof? Here are some related questions to explore:

1. What is the significance of proof by contradiction in logic?
2. How does disjunction (\( \vee \)) interact with implications?
3. Can you explain how the truth table for \( R \rightarrow S \) looks?
4. What happens if \( P \) is true in this scenario?
5. How can these logical principles apply to real-world situations?

**Tip:** When proving logical conclusions, looking for contradictions can be a powerful technique to show the necessity of a statement.

Explore a detailed step-by-step proof of the validity of R→S from the premises ~P∨Q, ~Q∨R, and R→S using propositional logic and proof by contradiction. This solution covers key logical concepts and theorems such as Modus Ponens and Modus Tollens. Ideal for undergraduate students studying logic.

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