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Blast into Math! Pure mathematics: the proof of the pudding is in the eating

1. What is the converse of the statement: if the product of two integers is positive, then they
are both positive?
2. What is the contrapositive of the statement from the preceding exercise?
3. For the statement from the preceding two exercises, which of the following are true: the
statement, its converse, its negation or its contrapositive?
4. What is the negation of the statement: every odd number is divisible by two.
5. What is the negation of the statement: Seattle is always cloudy.
6. What is the converse of the statement: if the sum of two integers is even, then they are both
even.

7. What is the contrapositive of the statement: when it rains, it pours.
8. Do some research into logic and symbols used in logic. Find the meaning of the symbols
“∀ ,” “∃ ”, and “∃!”
9. In one scene from Jim Hensen’s 1986 movie “Labyrinth,” starring Jennifer Connelly and
David Bowie, Jennifer’s character must choose between two doors: one door leads to certain
death, and the other door leads to the castle. Each of the doors has a talking door-knocker.
One always tells the truth, and the other always lies, but you don’t know which is which.

You may ask only one question to one of the door knockers, and it must be a question
whose answer is either “yes” or “no.” What question should you ask?
10. Do some research into the most commonly used endings of proofs, including q.e.d. and .
What do they mean? Where did they originate? Then, choose a symbol to end your proofs
and explain your choice. You may choose one of the commonly used endings, ♥, or your
own creation.

2.6 Examples and hints

If your goal is to learn mathematics, then you must make a strong individual effort to solve each of
the exercises before looking at the examples and hints. However, these problems are intended to be
challenging, so eventually, you will face a problem you cannot solve. When is it time to look at the
examples and hints or ask for help?

2.6.1 The Hint Rule
You have earned a hint when:

You have worked on the problem long enough so that you can state from memory both the problem
and the definition of every mathematical word in the problem completely and correctly.

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