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