Problem Set 1—Propositions

Purpose

This problem set reinforces your understanding of propositions (i.e., mathematical statements) and of proofs of propositions.

Background

This problem set is based on material in sections 1.1 and 1.2 of our textbook. We discussed (or will discuss) those sections in class on January 22 and 24.

Activity

Problem 1

For each of the following sentences, determine whether it is or is not a mathematical statement. For each that is a statement, decide whether it is true or false. Give a brief reason, although not a formal proof, why you classify it as true/false:

1. Every even number is a multiple of 2.
2. Winter is a nicer season in Geneseo than summer.
3. More than half of Earth’s atmosphere by volume is nitrogen.
4. x is a real number.
5. There is some real number that is greater than 7.
6. There is some real number, x, such that x > 7.
7. There is some real number, x, such that x > y.
8. If 1 < 2 then Engl 342 is a prerequisite for Math 239.
9. If Engl 342 is a prerequisite for Math 239 then 1 < 2.
10. For all real numbers x, if Engl 342 is a prerequisite for Math 239 then calculating 3x + 1 will cause a genie to appear and grant you three wishes.

Problem 2

A small variation on exercise 2a in section 1.2 of our textbook: give a formal proof that if x is an even integer and y is an even integer, then x + y is an even integer. Your proof should follow the rules for writing a formal proof, except that it doesn’t need to be typed. You may, but are not required to, write a know-show table or similar record of the informal thinking behind the proof; whether you write it down or not, be prepared to discuss your thinking during your grading appointment.

Problem 3

Give a formal proof that if n is an even integer, then 4n + 2 is an even integer. Once again, you should follow the rules for writing a formal proof except for typing it. You may but don’t have to record the informal thinking behind the proof, but I will probably ask you to discuss it during your grading appointment.

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.

I will use the following guidelines to grade this problem set:

• What I expect (8 points). Between your written answers and verbal explanations, I expect you to show that you understand (1) what constitutes a mathematical statement, (2) when simple mathematical statements are true and false, (3) when conditional statements are true and false, and (4) how to express formal mathematical proofs of conditional statements.
• Half of what I expect (4 points). Here some plausible, but not the only, examples of solutions that would meet half my expectations: you fully understand 2 of the 4 items listed as expected, and don’t understand the others at all, OR you partially understand all 4 items.
• Exceeding expectations (typically 1 point added to what you otherwise earn). Here are some plausible, but again not the only, examples of things that go beyond what I expect: particularly clearly written proofs, including proofs that are properly typeset OR particularly elegant — i.e., short yet clear — solutions, etc.