All numbered readings are from Beck and Geoghegan s The art of proof.

Transcription

1 MATH 301. Assigned readings and homework All numbered readings are from Beck and Geoghegan s The art of proof. Reading Jan 30, Feb 1: Chapters Feb 6, 8: Chapters Feb 13, 15: Chapters Feb 20, 22: Chapters Feb 22: Midterm 1 Feb 27, Mar 1: no reading Mar 6, 8: Chapters Mar 13: Chapters Mar 20, 22: Spring Break Mar 27, 29: Chapters Apr 3, 5: Chapters Apr 10: Chapter Apr 12: Midterm 2 Apr 17-19: Chapter 4.6 Apr 24-26: Chapter 5 May 1-3: Chapter 8 May 8-15: Chapter 13 1

2 Homework HW 1 (due Thursday, Feb. 1) Group homework. (In the groups formed in class. For this part, turn in only one writeup per group.) 1. (Many different ways of computing ) (a) On the first day of class, in what different ways did you compute the product ? (b) Abstracting these, write different algebraic expressions for the product a b c d. (c) Prove that the different expressions you wrote in part (b) are equal to each other. Your group should have at least two different expressions. 2. (Associativity of abcde.) (a) Write down all the ways of associating/grouping the product abcde into pairwise multiplications, without changing the order of the factors. For example, one such grouping is ((ab)c)(de). (b) Draw a diagram where you list all those groupings, and you connect two of them if one is obtained from the other by a single application of the associative law (xy)z = x(yz). (c) Conclude that all these different ways of computing abcde give the same answer. Individual homework. Prove the following statements, using only the Axioms of the integers. 1. If m is an integer, then ( m) + m = If m, n, p are integers then (m + n) + p = (n + p) + m. 3. If m, n, p, q are integers then (m(n + p))q = (qm)n + p(qm). 2

3 HW 2 (due Thursday, Feb. 8) Individual homework. Prove the following statements, using only the Axioms of the integers and the propositions we have proved in class. 1. If m, n, p are integers such that m + n = m + p, then n = p. 2. If a Z then a 0 = If m is an even integer then m 2 is an even integer. 4. If n is an integer such that n 3 = n, then n = 0, n = 1, or n = If a, b, c, d are integers then (a b)(c d) = (ac + bd) (ad + bc). Bonus. (a) Prove that there are 42 ways of associating the product abcdef into binary products, and that they all give the same result. (b) What about the product a 1 a 2 a n for larger n? 3

4 HW 3 (due Thursday, Feb. 15) Individual homework. Solve the following problems, using only the Axioms of the integers, Axiom 2.1 of the natural numbers and the propositions we have proved in class. You may use the usual properties of how +,, interact with =, but remember that we have only proved associativity of 5 terms. 1. (a) Prove that 1 is a natural number. (b) If we define 6 = 5 + 1, prove that 2 3 = (a) If x and y are integers and x y and x y, prove that x = y. (b) If a, b, c, d are positive and a > b and c > d, prove that ac > bd. 3. Prove that there is no integer a such that a 2 = Prove that 5 n 2 n is a multiple of 3 for every natural number n. 5. For which natural numbers k is it true that k 2 4k 3? Prove your answer. 4

5 HW 4 (due Thursday, Feb. 22) Solve the following problems, using only the Axioms of the integers and the definition of subtraction. 1. If x + y = x then y = (a) ( m)n = m( n) (b) m (n p) = p (n m) Solve the following problems using only the Axioms 2.1 of the natural numbers and the definition of the inequality symbols < and. You may use the usual properties of integers and the operations +,,, and equalities =. 3. (a) If a < b and c < d then a + c < b + d (b) If m < n and p 0 then mp np. In the following problems, you may use the usual properties of the integers and the natural numbers, the operations +,,, and equalities = and inequalities < and. 4. If n is a natural number, prove that n n. 5. If k is a natural number then k 3 k is a multiple of 3. 5

6 HW 5 (due Thursday, Mar. 8) Group homework. (In the groups formed in class. For this part, turn in only one writeup per group.) The tables below show the outcomes of the Paper Pool game for m n boards where 1 m, n 10. What do you notice? What do you wonder? 1. State, as precisely as possible, several observations and conjectures about the Paper Pool game that you think are true. You may include the observations sketched above, but please look for others. 2. State, as precisely as possible, questions about the Paper Pool game that you would like us to answer. Please state these both in terms of the table (e.g.: The entries along this line have this property ) and in terms of the game (e.g.: For m and n having this property, the outcome of the game is this one ). You don t have to prove anything (yet). Individual homework. For each one of the following statements, write the converse and the contrapositive. 1. If a month has fewer than 30 days, then it is February. 2. If p is a prime number, then p = 2 or p is odd. Next to each of the six resulting statements, indicate whether it is true or false. You don t have to prove your answers. Bonus. Prove some of your observations about the Paper Pool game. 6

7 HW 6 (due Thursday, Apr. 5) Individual homework. 1. Consider the sequence a n = 2 n 1 that we studied in class. Prove that { 1 for n = 1, a n = a n n 1 for n Prove that 2 4m 6 is a multiple of 10 for any m N. 3. Prove that for any n N. n k 2 = k=1 n(n + 1)(2n + 1) 6 4. Consider the sequence 2, 5, 10, 17, 26, 37, 50, 65,... (a) What do you think the next number of the sequence is? (b) Write down a recurrence relation for this sequence. (c) Write down an explicit formula for this sequence. (d) Assuming your answer to (b) is correct, prove that your answer to (c) is correct. 5. Prove that for any b Z and m, n Z 0. (b m ) n = b mn Bonus. Consider the sequence a n = 2 n 1 that we studied in class: 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, , , , , , , , , , , , , , (a) You noticed the pattern that the last digit in these numbers is: 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5,..., respectively. Can you prove this pattern? (Hint: See Problem 2.) 7

8 (b) You noticed the pattern that the number of terms of the sequence having n digits for n = 1, 2, 3,... is 3, 3, 4, 3, 3, 4, 3, 3,..., respectively. Can you formulate a precise statement about this? Can you prove it? 8

9 HW 7 (due Thursday, Apr. 19) Group homework. (In the groups formed in class. Turn in only one writeup per group.) 1. State precisely and prove the identity for Fibonacci numbers that your group discovered in class. Bonus. Why do the Fibonacci numbers live inside the fractions 1 89, , , , ,...? 9

10 HW 8 (due Thursday, Apr. 26) Individual homework. In questions 1, 2, 3, and Bonus 1 and 2, I will ask you to: a. List the possibilities for n = 1, 2, 3, 4, 5. b. Conjecture a formula for the number of possibilities for n. c. Prove your conjecture in part b. The answer to a. for n = 3 is given, to make sure you understand what the question is asking. 1. The ways of tiling a 2 n rectangle with 2 1 dominoes. 2. The subsets of {1, 2,..., n}., {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} 3. The ways of arranging the numbers 1, 2,..., n on a line. 123, 132, 213, 231, 312, If A, B and C are sets such that A = B and B = C, prove that A = C. 5. Solve Project 5.3 in the book. Bonus 1. The ways of writing n as an ordered sum of natural numbers. 3, 2 + 1, 1 + 2, Bonus 2. The ways of tiling a 2 n rectangle with 2 1 dominoes, if a tiling and its reverse are considered to be the same. 10

11 HW 8 (due Thursday, May 3) Individual homework. 1. Prove that {3(x + 3) : x N} = {3x : x Z} {x Z : x 10}. 2. Is each of the following statements true or false? If it is true, prove it. If it is false, give a counterexample. (a) If A B and C D, then A C B D. (b) If A B and C D, then A C B D. 3. (a) (5 points) Draw a Venn diagram illustrating the set equality A (B C) = (A B) (A C) (b) (10 points) Prove that equality. 4. (15 points) 11

12 HW 9 (due Tuesday, May 15) Individual homework. 1. Is the function f : R 0 R 0 given by f(x) = x injective? Is it surjective? Is it bijective? 2. If f : A B is bijective and g : B C is bijective, prove that f g : A C is bijective. 3. Prove that the set of natural numbers and the set of even natural numbers have the same cardinality. 12