P or Q , Q then R ,  P then M , negation M prove that primises conclusion is P and (P or Q) in discrete mathematics with using simple rules

To prove that the premises lead to the conclusion \( P \wedge (P \vee Q) \), we will go through the given premises step by step using basic logical rules such as Modus Ponens, Disjunction Elimination, and others. The premises are:

1. \( P \vee Q \)  (Premise 1)
2. \( Q \rightarrow R \)  (Premise 2)
3. \( P \rightarrow M \)  (Premise 3)
4. \( \neg M \)  (Premise 4)

We need to derive the conclusion \( P \wedge (P \vee Q) \).

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### Step-by-step derivation:

1. **Restate the premises:**
   - From Premise 1, \( P \vee Q \), we know that either \( P \) or \( Q \) is true.
   - Premise 2 tells us that if \( Q \) is true, then \( R \) is true.
   - Premise 3 states that if \( P \) is true, then \( M \) is true.
   - Premise 4 tells us that \( M \) is false (i.e., \( \neg M \)).

2. **Using \( \neg M \) and \( P \rightarrow M \):**
   - Premise 4 gives us \( \neg M \).
   - From Premise 3, \( P \rightarrow M \), using **Modus Tollens** (if \( P \rightarrow M \) and \( \neg M \), then \( \neg P \)), we can conclude:
     \[
     \neg P
     \]

3. **From \( \neg P \), deduce \( Q \):**
   - From \( P \vee Q \) (Premise 1), since \( \neg P \), we must conclude that \( Q \) is true (by **Disjunction Elimination**).
     \[
     Q
     \]

4. **From \( Q \), apply \( Q \rightarrow R \):**
   - Now that we know \( Q \) is true, we can use Premise 2, \( Q \rightarrow R \), and apply **Modus Ponens** to deduce:
     \[
     R
     \]

5. **Show the conclusion \( P \wedge (P \vee Q) \):**
   - Now, let’s check what we need for the conclusion \( P \wedge (P \vee Q) \).
   - The first part of the conclusion is \( P \), which we don’t have (we actually have \( \neg P \)). So, we cannot directly prove \( P \) is true.
   - However, for the second part, \( P \vee Q \), we already know from Premise 1 that \( P \vee Q \) is true.
   
   Therefore, this argument does not seem to support the conclusion \( P \wedge (P \vee Q) \) as is, because we derived \( \neg P \).

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### Conclusion:
The premises given do not directly lead to the conclusion \( P \wedge (P \vee Q) \). Instead, the steps show that \( P \) is false and \( Q \) is true, which conflicts with the desired conclusion. Therefore, the conclusion cannot be validly inferred from the premises as written.

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Do you want more details on any part of this explanation, or further clarification?

### Relative questions:
1. How does Modus Tollens apply in logical proofs?
2. Can the conclusion \( P \wedge (P \vee Q) \) be valid with other premises?
3. How do Disjunction Elimination and Modus Ponens differ in logical reasoning?
4. What are the common fallacies in reasoning with implications?
5. How does the negation of a premise affect the validity of a conclusion?

**Tip:** Always identify whether the premises lead to a direct contradiction before attempting a proof.

P or Q, Q then R, P then M, negation M prove that premises conclusion is P and (P or Q) in discrete mathematics with using simple rules

Explore how to prove that P and (P or Q) follow from the premises using fundamental rules in propositional logic, including Modus Ponens, Modus Tollens, and Disjunction Elimination. This problem is suitable for undergraduate students studying discrete mathematics.

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