Math 8 - Transitions to Higher Mathematics - Fall 2007

Professor: Alex Dugas my homepage
Office: 6510 South Hall
Office Hours: W 10-12, F 2-3

Prerequisites: Math 3B (with a grade of C or better).

Text: Larry Gerstein. Introduction to Mathematical Structures and Proofs. Springer-Verlag 1996.

Lecture: MWF 9:00 - 9:50 am. in Arts 1247 .

Section: You must sign up for and attend a discussion section as well.  The section times and locations for this course are as follows: 

• T TH   6:00-6:50,  Phelp  1444
• T TH   7:00-7:50,  Girv  2124

The GSI for this course is Tom Howard.  His office hours, in SH 6431L, are:

•  M 12 - 1
•  T,  TH  4 - 5 (note change below)

Announcements:

• Office Hours for Finals Week:  Mine: M 2 - 4, W 10 - 12.  Tom's: MTW 2 - 3.
  
• The Final Exam is next Thursday (12/13) from 8-11am.  It will be about twice as long as the midterm exams (probably 10 questions).   It will be cumulative, but still focus on the material covered since the last midterm.  (For example, the breakdown might be as follows: 1/6-1/4 on chapter 1, 1/4-1/3 on chapter 2 and 1/2 on chapters 3 and 4).  Here are some review problems.  Solutions to the Review Problems.  You should also review the problems on your old homeworks.
  
• Lecture notes from Wednesday's lecture (11/21) will be posted below.  If you will not make it to class on Wednesday, please look them over before next Monday's lecture.
  
• My office hours this week will be Monday (11/19) from 10-11 and 1-2, instead of Wednesday 10-12.
  
• Solutions to Midterm 2.  The mean was 23.
• The second Midterm Exam is next Wednesday (11/14) in lecture.  It will focus on sections 2.1, 2.3-2.8 and on parts of 3.1-3.3 from the text, and all lectures through this Friday's (11/9).  Here is a review sheet that includes some sample problems.  Solutions are now available.
• Solutions to the Midterm are now available.  The graded tests will be returned in section on Tuesday (10/23).
  
• The first Midterm Exam is next Friday (10/19) in lecture.  It will cover all of Chapter 1 and Section 2.1 from the text, and all lectures through Monday's (10/15).  Here are some review problems that are similar to the types of questions that might be on the test.  Solutions are now available.  Please note that problems 2a, 4b, 6-8 require knowledge of set notation, which we will cover on Monday and Wednesday.  Extra practice problems have been included on this topic, since there is no homework on it before the test.
  
• Each of you should come to my office hours at least once during the first three weeks of class to introduce yourself.  If the scheduled times do not fit your schedule, please email me or talk to me after class to arrange another time.
  

Course Timetable (subject to change)
Date	Topics	Reading	Homework	Due Date
F 9/28	Introduction to Proofs; Rules of Inference.	Ch. 1.1	1.1:  Ex. 1, 2a, 4	Th 10/4
M 10/1	Logical Connectives: Not, And, Or.  Truth Tables.	Ch. 1.2	1.3:  Ex. 1, 4
W 10/3	Conditional Propositions;    Biconditionals;  Converse.	Ch. 1.3	1.3: Ex. 2, 8, 11c,d, 18a,d	Th 10/11  worksheet solutions
F  10/5	Contrapositive; Logical  Equivalence; De Morgan's Laws.	Ch. 1.5	1.5: Ex. 3, 4, worksheet (#'s 1, 3)
M 10/8	Direct Proofs. Arithmetic Review Sheet.	Ch. 1.4	1.4: Ex. 2, 3a,b
W 10/10	Indirect Proofs.		Homework 3	Th 10/18 Solutions
F  10/12	Proof by Contradiction.
M 10/15	Sets.  Notation and Definitions.	Ch. 2.1	2.1: Ex. 2-5, 7	Th 10/25 Worksheet Solutions
W 10/17	Set Builder Notation.  Set Equality.	Ch. 2.1	worksheet
F  10/19	Midterm 1
M 10/22	Quantifier Notation.	Ch. 2.3	2.3: 1, 4a-c
W 10/24	Russel's Paradox.	(Ch. 2.2)	2.3:  2, 9	Th 11/1
F 10/26	Set Inclusion.  Subsets.	Ch. 2.4	2.4:  1, 4, 7a, 9
M 10/29	Set Operations.  Unions.   Intersections.  Complements.    Venn Diagrams.	Ch. 2.5	2.5:  1, 3, 6, 7
W 10/31	Families of Sets indexed by a set.   Intersection, Union of indexed families of sets.	Ch. 2.6	2.6: 2a-c, 6a-c	Th 11/8
F  11/2	Ordered Pairs.  Cartesian products.	Ch. 2.7, 2.8	2.7: 1, 2a,b, 8 2.8: 4a-c, e, 9
M 11/5	Class Canceled!
W 11/7	Functions.  Functions as sets of ordered pairs/graphs or as diagrams.  Composition.	3.1	3.1: 1, 2, 5, 7a,b	Tu 11/20
F 11/9	Injective/One-to-One Functions.   Surjective/Onto Functions.   Bijective Functions.	3.2	3.2:  2, 3, 9
M 11/12	Veteran's day: No Class
W 11/14	Midterm 2
F 11/16	Composition of Functions.  Inverse Functions.	3.3	3.3: 2, 7, 8, 11
M 11/19	Cardinality of Sets.  Equipotent Sets.	4.1	4.1: 3, 4a,b, 6	Th 11/29
W 11/21	Finite and Infinite Sets.  Pigeonhole principle.  Countable and Uncountable Sets.  Lecture Notes: pg. 1,  pg. 2,  pg. 3	4.2	4.2:  4,  5a,c
F 11/23	Thangksgiving Holiday: No Class
M 11/26	Countable and Uncountable Sets. (cont.)	4.3	4.3: 1, 4, 7,  (optional: 11)
W 11/28	Mathematical Induction.	2.10	2.10: 1, 3, 6	Th 12/6
F 11/30	Induction (cont.).  Cardinality of Power Sets.	2.10	2.10: 4, 9
M 12/3	Strong Induction.  Recursive Sequences.  Fibonacci Numbers.	2.10
W 12/5	Binomial Coefficients.  Pascal's Triangle (not on final)	(5.8)
F 12/7	Review.
Th 12/13	Final Exam - 8:00 - 11:00 am

Course Content and Goals:  We will cover most of Chapters 1-3 in the text and additional topics (most likely from Chapters 4 and 6) as time allows.  Initially, the focus will be primarily on understanding new notation and writing proofs involving easy math, but we will gradually move to more abstract concepts and problem solving strategies.  By the end of this course you should be able to

1. Understand and use mathematical notation relating to sets, functions and quantifiers.
2. Write clear, organized and logically correct proofs.
3. Think abstractly about fundamental mathematical concepts, including sets and functions.
4. Approach problems that you don't know how to answer, devise strategies to solve them and come up with proofs on your own.
   
   
   

Section/Classwork:  Attendance and participation in a discussion section is mandatory for this course.  In section, your T.A. will answer questions, review the homework, and cover additional examples.  But, more importantly, you will work on problems, often in small groups.  Your  work will occasionally be graded and contribute to your grade for the course.

Homework:  Homework exercises will be assigned in lecture and listed in the table above.  Homework assignments will be due once a week in section, usually on Thursdays.  You may work together on homework problems; however, you must write up your answers individually.  You must fully justify your answers in order to recieve full credit.  Late homeworks will not be accepted.  However, your lowest homework score will be automatically dropped.

Exams: There will be two in-class midterm exams on Friday October 19, 9:00 - 9:50 am and Wednesday November 14, 9:00 -9:50 am. Please arrive promptly.  The final exam will be Thursday December 13, 8:00 - 11:00 am.  The problems on the exams will be similar to ones from classwork and homeworks.   No make-up exams will be given, except in extaordinary circumstances.  If you have a serious conflict with any of these exams or miss one for any reason, it is your responsibility to notify me immediately so that other arrangements may be made.

Grades:  Grades will be computed from your scores on classwork, homeworks and exams as follows: Classwork/Section = 10%, Homework = 25%, Midterms = 15% each, Final = 35%.  No letter grades will be assigned until the end of the semester, and the exact grading scale will be curved relative to the difficulty of the exams.  However, a 90% or above will guarantee you at least an A, an 80% will be at least a B, and 70% will be at least a C.