1.	For each of the arguments below, define the propositions and state which rule of inference was used to conclude when the premise is True. If you cannot conclude based on the argument, state the reason.
Argument 	Propositions 	Rule of Inference

1. If 𝑛 is a real number such that 𝑛 > 3,
then 2𝑛 > 6
Suppose that 𝑛 >= 2, can you
conclude 2𝑛 > 6

2.If 𝑛 is a real number such that
𝑛 > 2, then 𝑛2 > 4. Suppose that
𝑛2 ≤ 4, then 𝑛 ≤ 2
not awarded.



Let's analyze each argument in terms of the propositions and rule of inference used.

### Argument 1: 
**Statement:** If \( n \) is a real number such that \( n > 3 \), then \( 2n > 6 \). Suppose that \( n \geq 2 \), can you conclude \( 2n > 6 \)?

#### Propositions:
- \( P \): \( n > 3 \)
- \( Q \): \( 2n > 6 \)
- Premise: \( P \rightarrow Q \)
- Additional assumption: \( n \geq 2 \)

#### Analysis:
- The premise is the implication \( P \rightarrow Q \), meaning if \( n > 3 \), then \( 2n > 6 \).
- However, the assumption you are given is \( n \geq 2 \), which includes values of \( n \) such as 2, 2.5, and 3 that do not satisfy \( n > 3 \).
- Since the premise only applies if \( n > 3 \), and the new assumption doesn't guarantee \( n > 3 \), **you cannot conclude \( 2n > 6 \)**. The condition for the premise to apply isn't met under the new assumption.

#### Conclusion:
- **Reason for no conclusion:** The assumption \( n \geq 2 \) does not imply \( n > 3 \), so you cannot apply the premise \( P \rightarrow Q \).
- **Rule of inference:** No valid inference rule can be applied here due to the invalid assumption.

---

### Argument 2:
**Statement:** If \( n \) is a real number such that \( n > 2 \), then \( n^2 > 4 \). Suppose that \( n^2 \leq 4 \), then \( n \leq 2 \).

#### Propositions:
- \( P \): \( n > 2 \)
- \( Q \): \( n^2 > 4 \)
- Premise: \( P \rightarrow Q \)
- New assumption: \( n^2 \leq 4 \)

#### Analysis:
- The premise states \( P \rightarrow Q \), meaning if \( n > 2 \), then \( n^2 > 4 \).
- The contrapositive of \( P \rightarrow Q \) is \( \neg Q \rightarrow \neg P \), i.e., if \( n^2 \leq 4 \), then \( n \leq 2 \).
- Since the assumption is \( n^2 \leq 4 \), you can apply the contrapositive of the original statement to conclude that \( n \leq 2 \).

#### Conclusion:
- **Valid conclusion:** Using the contrapositive of \( P \rightarrow Q \), you can conclude that \( n \leq 2 \) when \( n^2 \leq 4 \).
- **Rule of inference:** **Contrapositive rule**.

---

### Summary:

1. **First argument:**
   - **Propositions:** \( P \rightarrow Q \) where \( P \): \( n > 3 \), \( Q \): \( 2n > 6 \), but the assumption \( n \geq 2 \) does not satisfy \( P \), so no conclusion.
   - **Reason:** Invalid assumption.
   - **Inference rule:** None applicable.

2. **Second argument:**
   - **Propositions:** \( P \rightarrow Q \) where \( P \): \( n > 2 \), \( Q \): \( n^2 > 4 \), and the contrapositive \( \neg Q \rightarrow \neg P \) applies.
   - **Conclusion:** \( n \leq 2 \) (contrapositive rule).

---

Do you want further details or have any questions?

### Related Questions:
1. What is the difference between a contrapositive and converse in logic?
2. Can you provide an example of Modus Ponens?
3. How do you recognize when a rule of inference cannot be applied?
4. What is the contrapositive of \( P \rightarrow Q \)?
5. How can you formally prove the contrapositive inference rule?

### Tip:
Always check whether the premise or the contrapositive of a logical statement applies when given assumptions that seem to reverse the implication.

For each of the arguments below, define the propositions and state which rule of inference was used to conclude when the premise is True. If you cannot conclude based on the argument, state the reason.
Argument 1: If 𝑛 is a real number such that 𝑛 > 3, then 2𝑛 > 6. Suppose that 𝑛 >= 2, can you conclude 2𝑛 > 6?
Argument 2: If 𝑛 is a real number such that 𝑛 > 2, then 𝑛^2 > 4. Suppose that 𝑛^2 ≤ 4, then 𝑛 ≤ 2.

Explore logical arguments involving real numbers and inference rules like the contrapositive. This detailed breakdown helps students understand how to apply logical rules to mathematical propositions, suitable for Grades 11-12 or introductory college students.

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