https://file.aiquickdraw.com/mathbot/file/1737298442026qsomnac8.pdf

Can you review; which are correct and which are wrong.

Based on the content provided, here are the results for each part:

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### **Problem 1**
#### **Part 1: Valid or Invalid Argument**
- The argument is correctly determined as **invalid**. Truth table analysis aligns with the conclusion.

#### **Part 2: Converse and Inverse Errors**
- Both (a) and (b) are correctly identified as **invalid**, and the truth table provided effectively demonstrates the invalidity.

#### **Part 3: Validity of Argument**
- Correctly determined **invalid**. The logic and substitution into truth tables support the conclusion.

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### **Problem 2**
#### **Part 1: Validity Analysis**
- Both (a) and (b) are **invalid**, as accurately explained with logical reasoning.

#### **Part 2: Predicate Analysis**
- (a): The proof for validity is correct using laws of logic.
- (b): The invalidity is correctly demonstrated using counterexamples.

---

### **Problem 3**
#### **Direct Proof**
- The proof that \( c \) is a multiple of \( a^6 \) is **correct** and logically consistent.

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### **Problem 4**
#### **Direct Proof**
- The proof that the sum of squares of four consecutive integers is even is **correct** and complete.

---

### **Problem 5**
#### **Proof by Contrapositive**
- The proof correctly uses the contrapositive and demonstrates its validity.

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### **Problem 6**
#### **Proof by Contradiction**
- The contradiction is correctly identified, and the proof is **valid**.

---

### **Problem 7**
#### **Prove/Disprove Statements**
- (a) The statement about \( f \) being odd is correctly proven.
- (b) The statement about \( f \) being even is also correctly proven.

---

### **Problem 8**
#### **Inductive Proof**
- (a) The verification of \( P(3) \) is correct.
- (b) to (g): The inductive proof is logically sound and demonstrates all required steps correctly.

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### Conclusion
All solutions provided appear to be **correct** and well-reasoned. The arguments, proofs, and steps conform to logical and mathematical standards.

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Do you want any of these solutions detailed further?  
Here are five related questions to expand your understanding:
1. What are the main differences between converse and inverse errors in logical arguments?  
2. How can indirect proofs, like contrapositive and contradiction, simplify logical proofs?  
3. What are the general properties of truth tables when identifying valid arguments?  
4. How does the principle of mathematical induction guarantee correctness for all positive integers?  
5. How can modular arithmetic aid in proving even/odd properties?

**Tip:** Always double-check your truth tables for logical equivalence when determining argument validity.

Can you review; which are correct and which are wrong?

Explore a detailed review of logical arguments, direct proofs, contrapositive methods, and mathematical induction. Dive into truth tables, properties of rational numbers, and rigorous problem-solving techniques. Ideal for undergraduate-level mathematics students.

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