Question

1. Give a direct proof that the product of two odd integers is odd.

2. Give an indirect proof that if 2n 3 + 3n + 4 is odd, then n is odd.

3. Give a proof by contradiction that if 2n 3 + 3n + 4 is odd, then n is odd. Hint: Your proofs for problems 2 and 3 should be different even though your proving the same theorem.

4. Give a counter example to the proposition: Every positive integer which ends in 31 is a prime.

5. Give a proof by cases that min{s, t} + max{s, t} = s + t for any real numbers s and t. Hint: One of the cases you might use is s ≤ t or s < t. Depending on your choice, what would be the other case(s)?

Answer 1

Doubt in any step then comment below.. i will help you..

By rules and regulations we are allow to do only one question at a time..i do 2 questions...

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please thumbs up for this solution..thanks..

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