Autumn 2017
Math 220: Foundations of Mathematics

Homework

• Page 113: Number 7
• Page 119–120: Number 1 (a),(d),(i),(j),(n). Number 8
• Page 134: Numbers 2 and 3. (Number 2 can be done with either 9 cases, none hard, or with only 3 cases.).

• Page 102: Do number 6
• Page 107. Do numbers 1, 2, and 3. The first two are not proofs, but calculations.
• Page 113. Do number 1 (a)–(i), but only provide proofs for (a), (f), and (h). (The proofs can/should be short.)

• Page 88. Do numbers 1, 8 (Hint: let j be in I and show that A_j lies between the intersection and the union.)
• Page 102: Do numbers 3, 5
• In Sphereland, if shapes must be regular, what are the possible shapes ? (Look up Platonic, or regular solids.) In that case, is it possible for the number of sides to increase with each generation ? What do you think happens ? Write a couple of paragraphs about this and your explorations.

• Page 80. Do numbers 1(a),(c),(g), 2, 4, 8, 15, 25 (this is a bonus, as it may be harder)

• Page 73. Do number 1,2,3,4,5,7.

• 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.
• Starting on page 65 of our book, do the following exercises. Numbers 1, 2, and 4. Read, but do not do, number 5.

• On page 60 of our book, do numbers 2, 3, 6, 16, and 20. Do your best to write succinct proofs, ones that do not have many (or any) extraneous bits.

1. To get ready for the short answer question on the test, look at some of the videos about Klein Bottles: (You needn't stop with these.)
   • Klein bottle
   • How to make a Klien bottle in three dimensions
   • Cutting a Klein bottle in half
2. 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 using the contrapositive.
3. 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.)
4. Please do the following exercises from Section 2.2:   4 (4 might rely on the answer to 1, so you can use it), 8, 9, 11.
5. Do Exercises 2.4, Number 1 (a), (c), (e), (g), (i), (k).

1. Read the Appendix: Writing Mathematics.
2. From Exercises 2.1 do:
   • 3. (This uses Proposition 2.1.9.)
   • 5. (a), (b), and (e).
   • 15.
   • 17. (This will require working backwards.)
   • 20.
3. 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.)