Spring 2019
Math 220: Foundations of Mathematics

Homework

1. Page 144: Numbers 3, 4, 6, 7.
2. Page 147: Numbers 2, 5, 7, 11.
3. Page 152: Number 1.

1. Page 134: Numbers 2, 3, and 5.
2. Page 140: Numbers 1, 3, and 6.

1. Page 113. Do number 1 (a)–(i), but only provide proofs for (a), (f), and (h). (The proofs can/should be short.) Also do numbers 4 and 7.
2. Page 119–120: Number 1 (a),(d),(i),(j),(n). Number 8.
3. Page 128. Numbers 1, 4, 5.

1. Page 88. Do numbers 1, 8 (Hint: let j be in I and show that A_j lies between the intersection and the union.)
2. Page 102. Do numbers 1(a)(d)(e), 2, 5, 6
3. Page 107. Do numbers 1, 2, and 3.

1. Read Flatland, probably twice. Before re-reading, peruse this study guide.
2. Section 4.2, numbers 4, 5(a) and (e), 8, 12, 13, 15 and 21 (c) on pages 80 and 81.
3. On page 61 of our book, do numbers 18 and 19. These are important facts for algebra and combinatorics, and it is good to work through a proof of them, now that you have the tools to do so. Also do number 20.

1. A tiny bit on induction. From page 60 in our book, do numbers 3 and 12.
2. Page 73. Do numbers 1,2,3,4,5,7.
3. Page 80. Do numbers 1(a),(c),(g)

1. To get ready for the short answer question on the test, look at some of the videos about Klein Bottles: (Do look up a couple more sources, too, and above all, enjoy them.)
   • Klein bottle
   • How to make a Klien bottle in three dimensions
   • Cutting a Klein bottle in half
   Hand in one paragraph about a particular aspect of Klein bottles, your choice, but do not write in generalities.
   The test (on 14 February) will have a short answer question, in which you will be asked to write a couple of paragraphs on Klein bottles, addressing one of several prompts.
2. On page 43, Exercises 2.2 numbers 1, 7, 8, and 9. Think about #12, it is an insight of Archimedes.
3. Page 51, Exercises 2.4, do #1, all of it. This will help you understand mixed quantifiers.

1. 15, from Exercises 2.1 on page 37.
2. 17, from Exercises 2.1 on page 37. (This will require working backwards.)
3. 20, from Exercises 2.1 on page 37.
4. Prove the following statement by contradiction (reductio ad absurdum): For all integers n, if n² is odd, then n is odd. It is worthwhile to compare this to the proof in class using the contrapositive.
5. Prove the following statement by contradiction (reductio ad absurdum): For all real numbers a and b with b≥0, if a²≥b, then either a≥√ b  or a≤-√ b . (The √ b    is a plain .html way to express the square root of b.)
6. Bonus: The sequence of integers n²+ n +41 is rich in primes for n>0. How rich? For how many integers n less than 100 is this prime? How about less than 1000? What about stupendously large numbers? You should write a program to test this and include it with your answer. (Sottile's program is 3 lines long in Maple.)

1. Read the Appendix: Writing Mathematics.
2. On page 23 of our book, the Basic Properties of the Integers are found, and on page 27, there are the basic properties of the real numbers. A not-so-close reading shows that these are near the same. What is different? Which of the properties for one hold for the other, and vice-versa.
   There are two others, properties, the well-ordering principle for the positive integers and the completeness principle for the real numbers, which are dramatically different. We will cover at least the first of these in our course, the second occurs in adavanced calculus.
3. From Exercises 1.2 do:
   Numbers 1 and 2 (these were from last homework; I did not get far enough before assigning them for you to so them.)
4. From Exercises 2.1 do:
   • 3. (This uses Proposition 2.1.9.)
   • 5. (a), (b), and (e).
   • 8.