Department of Mathematics

Transitions, MTH 299-03

Tentative Schedule and Homework Assignments

Chapter numbers refer to:

[H] Kevin Houston, How to Think Like a Mathematician
[B] Matthias Beck, The Art of Proof (available online)

L Date Chapter and Topic HW Comment
1 8/28 [H2,3,4] Reading and writing mathematics H3.2(ii), due 8/30 Supplement
R 8/29 Recitation
2 8/30 [H1] Sets and functions H1.10,20(ii),23(ii) due 9/4 You can start working on Problem 34, as you should be able to do a large part of it. We have not discussed diffference of sets yet.
9/2 Labor Day - no class
3 9/4 [H1] Sets and functions; examples, models, enumeration H1.33(i-ii),34(i-iv), due 9/6
HW4, due 9/9 		Supplement
R 9/5 Recitation
4 9/6 [H30] [B13.1] Injective, surjective, bijective functions H30.13, 30.16, 30.28(i,iii,v), due 9/11 Note that the domain for 30.16 should be [-pi/2,-pi/2].
For 30.28(i) give a brief answer to all but provide rigorous proof on whether the function is injective and/or surjective only for (a) and (e).
5 9/9 Combinatorial enumeration and bijections HW5, due 9/13 Supplement
Problems 3, 4, 5 are NOT mandatory. Only turn in problems 1 and 2 with detailed and clear solutions.
6 9/11 [B13.2-3] Cardinality, countability, Cantor's arguments HW6, due 9/18
Proof Essay, due 9/18
R 9/12 Recitation
7 9/13 Cardinality, Cantor's argument - continued Problem 2 from Quiz 3, due 9/16 Look at your lecture notes and recitation notes and write a detailed, rigorous proof for Problem 2.
8 9/16 [H6] Making a statement Reading: Ch 6, pp 53-62
H6.2, 6.11(ii-d,e,f),(iii),(iv),(v-a,d) due 9/20 		For 6.2 explain what is the reason a sentence is or is not a statement. (One sentence of explanation per problem should be enough.)
You can consruct one "long" table for (ii) and another one for (iv).
9 9/18 [H7] [B3.2] Implications Reading: Ch 7, pp 63-67
H7.2, 7.7(i-iv), due 9/23
R 9/19 Recitation
10 9/20 [H8] [B3.2] More on implications Reading: Ch 8, pp 69-74
H8.10, 8.13(i-iv), due 9/25
11 9/23 [H9] [B3.3] Converse and equivalence Reading: Ch 9, pp 75-78
H9.3,9.9(iii,iv-a,b,c), Write a rigorous proof of the foollowing:"A natural number, n, is odd if and only if its square is odd.", due 9/27
9/23 Last day to withdraw with refund
12 9/25 [H10] [B3.1] Quantifiers: for all, there exists Reading: Ch 10, pp 80-83
10.11, due 9/30
R 9/26 Recitation
13 9/27 [H11,12] [B3.3] Negation of quantifiers Reading: Ch 11 and 12, pp 84-95
11.12(i), (iii), (iv), (v-a,c), 12.4, due 10/2		 Give a rigorous proof for 11.12(v-a,c). In (a) you will need to find a particular n that satifies the conditions and show that it satisfies them, in (c) you need to find an N, which will depend on epsilon, and show it satisfies the desired condition for all n>N.
14 9/30 Review Review worksheet
15 10/2 Review
M 10/3 Midterm Exam
16 10/4 [H2,14,15,16] Definitions, Theorems, Axioms Homework on axioms, due 10/9 Read sections 14, 15 and 16
17 10/7 [H2,14,15,16] Definitions, Theorems, Axioms Homework on Chapter 16, due 10/11
18 10/9 [H23] Proof by Contradiction 23.7(ii,iii,iv), 23.8(x), due 10/14 Read section 23
R 10/10 Recitation
19 10/11 [H20] Proof by direct argument Proof Essay 2 and
20.14(i, iv-b, v-c), due 10/16
20 10/14 [H21,22] Common Mistakes; Proof by Cases 22.10 (v, vii), due 10/18 Read sections 21 and 22
21 10/16 [H26] Proof of contrapositive 26.7(iv)
2) If a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10.
3) Let be an integer. If x^2+6x+5 is even, then x is odd.
due 10/21 		Read section 26
Lecture Notes
10/16 Last day to withdraw with no grade reported
R 10/17 Recitation
22 10/18 [H24,25] Proof by induction 24.10(i,ii,iii) and
Revision of Proof Essay 2, due 10/23		 Read section 24
Lecture Notes
23 10/21 [H24,25] Proof by induction 25.7(ii), due 10/25 Read section 25
24 10/23 Exercises on choosing proof method Provide a proof for Problem 13 from the Lecture Notes, due 10/28
R 10/24 Recitation
25 10/25 Example: divisibility of integers, infinity of primes Homework, due 10/30 Lecture Notes
26 10/28 [H27,28] Common divisor gcd(a,b), Euclidean algorithm Prove the criterion for divisibility by 4
and
Answer the questions at the end of the Lecture Notes handout 		Lecture Notes
27 10/30 [H28] Euclidean algorithm, unique factorization theorem 28.19(viii) (Use Euclid's Lemma), due 11/4
Make sure you are prepared to further discuss the proof of the Division Lemma on Friday 		Lecture Notes
Supplement
R 10/31 Recitation
28 11/1 [H29,31] [B6.1,3] Modular arithmetic, equivalence classes Midterm 2 Review + Solutions
Work on these sample problems and example problems from previous lecture notes.
Bring questions you would like us to work together on to class. 		Lecture Notes
29 11/4 [H29] [B6.3, App B] Extra: finite fields, cryptography
30 11/6 Review
M 11/7 Midterm Exam
31 11/8 Modular Arithmetic Homework, due 11/13 Lecture Notes
32 11/11 Modular Arithmetic and Equivalence Classes Homework, due 11/15
Read Supplement
33 11/13 [B8.3-4] Real numbers, upper and lower bounds, completeness axiom Read Chapter 8, "The Art of Proof"
Homework, due 11/18 		Lecture Notes
R 11/14 Recitation
34 11/15 More on axioms of real numers Supplement on axioms
Homework, due 11/20
35 11/18 >[B10.4] Limits of Sequences Homework, due 11/22 Lecture Notes
36 11/20 [B10.4] Limits of Sequences Homework, due 11/25 Lecture Notes
R 11/21 Recitation
37 11/22 Properties of Limits Homework, due 11/27
38 11/25 Series as limits of partial sums Homework, due 12/4
39 11/27 Series Homework, due 12/4 Lecture Notes
11/28 Thanksgiving, no recitation
11/29 Thanksgiving, no class
40 12/2 Review Review for material after exam 2
41 12/4 Review Additional Review Problems Solutions (the solutions start on page 6)
R 12/5 Recitation
42 12/6 Review
E 12/9 Final Exam 10:00am-noon Brody Hall 112 Make Up Final Registration Form The make up final is on 12/10 10am-noon in room N130, Business College Complex