Problem Set 1 — Statements

Purpose

This problem set develops your ability to recognize mathematical statements, including conditional statements, and to determine informally when a statement is true and when it is false. The problem set thus addresses the following learning outcome:

• Outcome 6, unravel abstract definitions, create intuition-forming examples or counterexamples, and prove conjectures

Background

This exercise is based on section 1.1 of our textbook. We discussed, or will discuss, that material in classes on February 5 and February 8.

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. Beethoven’s Ninth Symphony is the best piece of music ever written.
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 \(x\) is a real number and \(x > 10\), then \(x+1 > 11\).

Problem 2

Determine whether each of the following conditional statements is true or false. Briefly explain, although not necessarily in a formal proof, why each is true/false:

1. If \(1 < 2\) then Engl 342 is a prerequisite for Math 239.
2. If \(1 < 2\) then Math 222 is a prerequisite for Math 239.
3. If Engl 342 is a prerequisite for Math 239 then \(1 < 2\).
4. 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.
5. If New Years Day of 2099 falls on a Sunday, then Prof. Baldwin sometimes teaches Math 239.

Follow-Up

I will grade this exercise during one of your weekly individual meetings with me. That meeting should happen on or before the “Grade By” date above. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.