# Solutions to Problem Set 7

Math 310-01/02

9-20-2017

1. (a) Prove that if n is even, then is even.

(b) Give a specific counterexample to show that the converse is false.

(a) Let n be an even integer. Then for some . So

Hence, is even.

(b) The converse is: "If is even, then n is even." However, if , then , which is even, while n is odd. Thus, the converse is false.

2. Prove that if , then .

I'll prove the contrapositive: If , then . Suppose that . Then and , so

Therefore, .

3. (a) Show that if , then or .

(b) Show that it's not true that if or , then .

(a)

Therefore, or .

(b) Plugging into gives , which is true. The answer checks.

Plugging into gives , which doesn't make sense --- the log of a negative number is undefined. The answer doesn't check.

4. Premises: .

Prove: .

We are what we think. All that we are arises with our thoughts. With our thoughts we make the world. - The Buddha

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