Directions: Prove the following propositions by using the kind of proof indicated in each number. Use outline form for numbers 1 and 2. while paragraph form for numbers 3 and 4

Question

Directions: Prove the following propositions by using the kind of proof indicated in each number. Use outline form for numbers 1 and 2. while paragraph form for numbers 3 and 4

1. If a is an odd integer, then a²+3a+5 is odd. (Direct Proof)

2. Suppose x, y € Z. If x is even, then xy is even (Indirect Contraposition Proof)

3. For all real numbers a and b, if a² = b² , then a = b (Proof by Counter Example)

4. For all integers n. if n³ + 5 is odd then n is even. (Proof by Contradiction)

Answer 1

Hi Kevin,

This is a great question that I have seen come up on many tests/exams. If this answer is helpful, I would greatly appreciate your feedback as I am new to the platform, yet an experienced expert in mathematics!

Direct Proof

Statement: If a is an odd integer, then a^2 + 3a + 5 is odd.

Proof: Assume a is odd. This means a = 2m + 1 for some integer m. Substitute this into a^2 + 3a + 5 to get (2m + 1)^2 + 3(2m + 1) + 5. Simplifying, we get 4m^2 + 10m + 9, which can be expressed as 2(2m^2 + 5m + 4) + 1. This is of the form 2k + 1, proving that a^2 + 3a + 5 is odd.

Indirect Contraposition Proof

Statement: If x is even, then xy is even.

Proof: Assume xy is not even, meaning it's odd. This implies xy = 2n + 1 for some integer n. If x was even, then x = 2m for some integer m, making xy = 2mn, which is even. This is a contradiction, proving x cannot be even if xy is odd.

Proof by Counter Example

Statement: For all real numbers a and b, if a^2 = b^2, then a = b.

Counter Example: Consider a = 1 and b = -1. Here, a^2 = b^2, but a is not equal to b. Therefore, the statement is false.

Proof by Contradiction

Statement: For all integers n, if n^3 + 5 is odd, then n is even.

Proof: Assume n^3 + 5 is odd and n is odd. This means n = 2m + 1 for some integer m. Substitute this into n^3 + 5, which becomes (2m + 1)^3 + 5. Simplifying, we get 8m^3 + 12m^2 + 6m + 6. This is even, contradicting our original assumption that n^3 + 5 is odd. Therefore, n must be even if n^3 + 5 is odd.

Thank you,

Benjamin M.