State the proof strategy you use **Mathematical Properties and Proofs**In the study of mathematics, certain properties and relationships between numbers are fundamental. Here we explore some of these properties with proofs and explanations:1. **Even Products of Integers**: - **Statement**: Let $ m $ and $ n $ be integers. Show that if $ mn $ is even, then $ m $ is even or $ n $ is even. - **Explanation**: This property suggests that for the product of two integers to be even, at least one of the integers must be even. This can be proven using the definitions of even and odd numbers and the distributive properties of multiplication.2. **Rational and Irrational Numbers**: - **Statement**: If $ r $ is a nonzero rational number and $ p $ is an irrational number, then $ rp $ is irrational. - **Explanation**: This proposition highlights that the product of a nonzero rational number and an irrational number remains irrational. This is a key aspect in understanding the behavior of rational and irrational numbers under multiplication.3. **Maximum Property of Real Numbers**: - **Statement**: For all real numbers $ a $, $ b $, and $ c $, $ \max(a, \max(b, c)) = \max(\max(a, b), c) $. - **Explanation**: This property deals with the maximum function and demonstrates its associative nature. It states that finding the maximum of three numbers is consistent regardless of the grouping of operations.4. **Rational Numbers and Irrationals**: - **Statement**: If $ a $ and $ b $ are two distinct rational numbers, then there exists an irrational number between them. - **Explanation**: This theorem signifies the density of irrational numbers within the real number line, ensuring that between any two distinct rational numbers, one can always find an irrational number.5. **Sum of Even and Odd Integers**: - **Statement**: If $ m + n $ and $ n + p $ are even integers where $ m $, $ n $, $ p $ are integers, then $ m + p $ is even. - **Explanation**: This result shows a relationship between sums of integers being even. It can be inferred using basic properties of even and odd numbers and their arithmetic behavior.6.

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Description: State the proof strategy you use **Mathematical Properties and Proofs**In the study of mathematics, certain properties and relationships between numbers are fundamental. Here we explore some of these properties with proofs and explanations:1. **Even

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(a) Let m and n be integers. Show that if mn is even, then m is even or n is even. (b) If r is a nonzero rational number and p is an irrational number, then rp is irrational. (c) For all real numbers a, b, and c, max(a, max(b, c)) = max(max(a, b), c).

(a) Let m and n be integers. Show that if mn is even, then m is even or n is even. (b) If r is a nonzero rational number and p is an irrational number, then rp is irrational. (c) For all real numbers a, b, and c, max(a, max(b, c)) = max(max(a, b), c).

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Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

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State the proof strategy you use

**Mathematical Properties and Proofs**

In the study of mathematics, certain properties and relationships between numbers are fundamental. Here we explore some of these properties with proofs and explanations:

1. **Even Products of Integers**:
- **Statement**: Let $ m $ and $ n $ be integers. Show that if $ mn $ is even, then $ m $ is even or $ n $ is even.
- **Explanation**: This property suggests that for the product of two integers to be even, at least one of the integers must be even. This can be proven using the definitions of even and odd numbers and the distributive properties of multiplication.

2. **Rational and Irrational Numbers**:
- **Statement**: If $ r $ is a nonzero rational number and $ p $ is an irrational number, then $ rp $ is irrational.
- **Explanation**: This proposition highlights that the product of a nonzero rational number and an irrational number remains irrational. This is a key aspect in understanding the behavior of rational and irrational numbers under multiplication.

3. **Maximum Property of Real Numbers**:
- **Statement**: For all real numbers $ a $, $ b $, and $ c $, $ \max(a, \max(b, c)) = \max(\max(a, b), c) $.
- **Explanation**: This property deals with the maximum function and demonstrates its associative nature. It states that finding the maximum of three numbers is consistent regardless of the grouping of operations.

4. **Rational Numbers and Irrationals**:
- **Statement**: If $ a $ and $ b $ are two distinct rational numbers, then there exists an irrational number between them.
- **Explanation**: This theorem signifies the density of irrational numbers within the real number line, ensuring that between any two distinct rational numbers, one can always find an irrational number.

5. **Sum of Even and Odd Integers**:
- **Statement**: If $ m + n $ and $ n + p $ are even integers where $ m $, $ n $, $ p $ are integers, then $ m + p $ is even.
- **Explanation**: This result shows a relationship between sums of integers being even. It can be inferred using basic properties of even and odd numbers and their arithmetic behavior.

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