| Thirteenth homework :
(Due Monday, 2 December 2019 at the start of class) Chapter 9 in the 3rd edition is Chapter 10 in the fourth edition From Chapter 9/10 do: 3, 6, 12(b) and 12(e), 16, 23, 24, 32, 38, 40, 42(b)(d). Let Z be the integers and Z* be the nonzero integers. Define a relation Q on Z×Z* by (a,b)Q(c,d) if ad = bc. Prove that Q is reflexive, symmetric, and transitive. What is the set of Q-equivalence classes? Try to prove your assertion (this may be a bit challenging). |
| Twelfth homework :
(Due Wednesday, 20 November 2019 at the start of class) Chapter 8 in the 3rd edition is Chapter 9 in the fourth edition From Chapter 8/9 do problems 36, 38, 42, 44, 46, 52, 54, 56, 58, 60, 62, 70. FYI: from two weeks ago, here is the maple file and output from Sotttile's computation of primes in n2+ n +41. Fake_Prime_Seq.maple |
| Eleventh homework :
(Due Monday, 11 November 2019 at the start of class) Let fn be the nth Fibonacci number. This satisfies the recursion f1= f2=1, and for n>1, we have fn+1= fn + fn-1. (1) Give a proof by induction that for all natural numbers n, f3n is even. (2) Give a proof by induction that for all natural numbers n, f5n is divisible by five. (3) Guess a formula for f1+ f2 + ... + fn, the sum of the first n Fibonacci numbers, and prove it. (4) Guess a formula for f12+ f22 + ... + fn2, the sum of squares of the first n Fibonacci numbers, and prove it. Consider the relation < (lexicographic order) defined on N×N as follows: (a,b)<(c,d) if and only if either a<c or a=c and b<d. (5) What properties does this relation have ? (reflexive, symmetric, asymmetric, transitive, ...) (6) Show that for every two pairs (a,b) and (c,d) of natral numbers, exactly one of the following holds: (a,b)<(c,d) or (a,b)=(c,d) or (c,d)<(a,b) (trichotomy). (7) Show that every nonempty subset of N×N has a <-least element. Read/skim Chapter 7 and do 7.32, 7.34, 7.38 (from 3rd edition). From Chaper 8 do 8.18, 8.22 (also from 3rd edition). Bonus: Let r be a natural number. Can you generalize (1) and (2) to a congruence that fan satisfies which holds for every n in N? (With proof.) Hand this in on a separate sheet of paper to Sottile. |
| Tenth homework :
(Due Wednesday, 6 Novemober 2019 at the start of class) From Chapter 6 do problems 24, 36, 40, 45, 61. These are from Edition 3, please post the equivalent for Edition 4 on Piazza. Write perfect proofs! 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.) Hand this in on a separate sheet of paper to Sottile. |
| Ninth homework :
(Due Wednesday, 23 October 2019 at the start of class) From Chapter 6 do problems 2, 6, 10, 14, 15, 18. These are from Edition 3, please post difference to Edition 4 on Piazza. Write perfect proofs! |
| Eighth homework : (Due Monday, 14 October 2019 at the start of class) From Chapter 4 do problems 54, 59, 64, 66(a). From Chapter 5 do problems 4, 8, 16, 18, 28, 34. In edition 3, these are 4, 8, 14, 16, 26, and 32. |
| Seventh homework : (Due Monday, 7 October 2019 at the start of class) From Chapter 4 do problems 30, 34 (leftover from last week). Also, numbers 38 and 39, and 40 (and then prove that the three sets form a partition of the union of A and B), as well as 42, 44, and 46. |
| Sixth homework : (Due Monday, 30 September 2019 at the start of class) From Chapter 4 do problems 2, 6, 16, 18, 27, 30, 34. (I do not know if these differ in the 3rd edition, we can have a Piazza discussion). |
| Fifth homework : (Due Monday, 23 September 2019 at the start of class) This is intended to help you study for the exam, but it is not a review. From Chapter 2 do problem 101 (a) and (b) in the fourth edition, which is number 99 (a) and (b) in the third edition From Chapter 2 do problem 102 in the fourth edition, which is number 100 in the third edition Exercises from Chapter 3: 18, 21, 24, 28, 29, 42 (38 in 3rd edition), 70 (66 in 3rd edition). |
| Fourth homework : (Due Monday, 16 September 2019 at the start of class) Do the problems found on this page. Do the following exercises from Chapter 2 in 4th edition: Numbers 49 (this is the same in both editions). Numbers 54, 60, 68, 70, 72, 82, 94 (I think that you must subtract 2 from these for the third edition. Please discuss this on Piazza, I will type in what I expect when you prompt me; I only have the 4th edition with me at the Casa Mathematica Oaxaca.) Do the exercises in Chapter 3 in the 4th edition: 2, 6, 10, 12. |
| Third homework : (Due Monday, 9 September 2019 at the start of class) Do the following exercises from Chapter 2 in 4th edition: Numbers 2, 4, 14, 16, 18, 20, 22, 26, 27 (explain your reasoning), 34, 36, 44, 46, 48. These are the same in both editions. |
| Second homework : (Due Monday, 2 September 2019 at the start of class) Do the following exercises from Chapter 1 in 4th edition: Numbers 4, 10, 12, 20, 24, 26, 36, 46, 48, 52, 66, 92. For the 3rd edition, numbers 46 and 92 will appear on Piazza, and 48, 52, 66 from the 4th edition becomes 46, 50, 64 in the third edition. |
| First homework : (Due second day of class, Wednesday, 28 August 2019) (1) Read Chapter 0 Communicating Mathematics. (2) Watch several Numberphile videos. (3) Write at a paragraph or two on your favorite one; indicating what it was about and what you learned. (4) Write a short recommendation about a second one. Hand in a printed copy of (3) and (4), and include the names of the other Numberphile videos you watched for this assignment (along with your name, etc.). |
| Zeroeth Assignment: Read the course web page, and send Frank an email that you have read and understood the course descriptions and policies. This includes signing up on Piazza. Please also answer the following questions: (1) Why are you taking this course? (2) What do you hope to get out of this course? (3) Is there anything else that you want to tell me (that is relevant to the course)? |