Class Hours: MWF 12:20 - 1:10 in MM207.
Text:
An Introduction to Mathematical Thinking
by William J. Gilbert and Scott A. Vanstone,
Pearson Prentice Hall 2005,
ISBN: 0-13-184868-2, QA10.G55 2005
Additional reading:
1)
What Is Mathematics? An Elementary Approach to Ideas and Methods
by Richard Courant (Author), Herbert Robbins (Author), Ian Stewart (Editor)
2) Bridge to Abstract Mathematics: Mathematical Proof and Structures by Ronald P. Morash
3) Discrete Mathematics and Its Applications
by Kenneth H. Rosen
4)
Introduction to Mathematical Thinking
by Keith Devlin
Description:
MTH230 Introduction to Abstract Mathematics is a one semester
proof oriented course for Mathematics majors. Selected topics covered: sets, logic (Propositional Calculus), elements of combinatorics (permutations, combinations, the binomial formula), the natural numbers (mathematical induction), the integers (arithmetic, divisibility), relations, functions, the rational, real and complex numbers, and cardinality. The course is an essential prerequisite for MTH433 and/or MTH533-534.
Prerequisite or corequisite:
MTH 162 or 172.
Grading policy:
Homework (20%), Exam 1 (20%) on Monday 3/4, Exam 2 (20%) on 4/... and Final (40%) on 5/....
General rules / information:
- check this page for updates.
- make sure that you have access to the UM listserv (email list via Blackboard) for this class
- you must show your work in order to obtain credit.
This page will be updated as we progress through the material.
Homework and Notes
Sets - review,
Sets and Logic,
Logical Equivalences.
Homework 1 - due Friday 22
Exercise Set 1: 5, 12, 24, 34, 38, 46, 50, 56, 60, 66.
Solutions
Problem Set 1: 76, 78, 80, 82.
Solutions
Review of set theory [not collected]
Recommended practice from
handout:
18, 24, 25, 29, 30, 48 Handout
Solutions
Homework 2 - due Friday 1/29
Do the exercises from the handout.
Solutions
Homework 3 - due Friday 2/5 - only problems with *
are graded.
Exercise Set 4: 6*, 7, 16*, 22*, 28*, 30, 32*, 33, 42*, 45.
Problem Set 4: 46*, 48, 50, 63*, 66*, 68*, 70, 74*, 75*.
Solutions
Homework 4 - due Wednesday 2/24 - only problems with * are graded.
Exercise Set 2: 2, 8*, 16*, 18, 22, 27, 34*, 44*, 58*, 71.
Problem Set 2: 76*, 82, 88, 89, 92*, 93, 94*, 98*, 100*, 106.
Solutions
Problem 95
Problem 88
Divisibility Criteria
Midterm 1 - 2/15
Covers all the material covered in class. Practice: homework, posted solutions, examples in the text.
Chapter 0 (Sets):
Sets, inclusion, operations with sets, complements, universal set, De Morgan's laws.
Chapter 1 (Logic):
Propositions, quantifiers, negation, implication (conditional), equivalence (bi-conditional),
contrapositive, truth tables, negation of predicates, use of counterexamples.
Chapter 4 (Induction and the Binomial Theorem):
Induction, recurrence, Fibonacci numbers, binomial formula and applications,
inequalities proven by induction.
Chapter 2 (Integers):
Definition of divisibility, division and Euclidean algorithms, gcd, primes, gcd with primes,
numbers in different bases, diophantine equations.
Solutions Midterm 1 - sent by Blackboard email
Homework 5 - due Friday 3/4
Do problems
2, 3, 6, 7, 10, 13 from the handout:
Relations on a set - notes and problem list
Solutions
Homework 6A - due Friday 3/18 - only problems with * are graded.
Exercise Set 3: 2, 6*, 10, 20*, 23, 24, 30*, 32, 40*, 44, 55*.
Solutions to 55, 55 modified, and 62
Homework 6B - due Friday 3/25 - only problems with * are graded.
Problem Set 3: 60*, 62*, 65, 71*, 82*, 84*, 94*, 103*.
Solutions and hints for Chapter 3
Homework 7 - discussed in class, not collected, but included on Midterm 2
Exercise Set 5: 4*, 6*, 10*, 32*.
Problem Set 5: 33*, 35*, 37*, omit problems with change of base, 51*
Solutions
Midterm 2 - Friday 4/1
Covers all the material covered in class since Midterm 1.
Practice: homework, posted solutions, examples in the text.
Chapter 2:
*** Linear diophantine equations from Chapter 2 are sometimes useful in Chapter 3.
Like in example 3.56 p 71.
Chapter 3 (Modular arithmetic):
Definition of congruence classes, tests for divisibility, last digits of a number in base 10,
addition and multiplication tables,
solving equations in Z_m, Fermat's little theorem,
***Chinese remainders theorem.
Chapter on Relations (see handout and Homework 6):
Relations, Equivalence relations, order relations, reflexivity,
symmetry and antisymmetry, transitivity, examples.
***Notice that Chapter 5: 1, 33, 34 are also on equivalence relations.
Chapter 5 (real and rational numbers, decimal representation):
Definition of rational numbers, proof that a number is irrational,
decimal representation, and exercises done in class - see above Homework 7, which is not
required to turn in, but discussed in class.
Solutions Midterm 2
Homework 8 - due [extended to] Wednesday 4/13.
Exercise Set 6: 2*, 4*, 8*, 12*, 15*, 28, 34*, 36*, 40, 41*, 43*, 45 [omitted, but covered for the final].
Solutions
Homework 9 - due Friday 4/22.
Exercise Set 6: 71*, 78*, 80*.
Problem Set 6: 81*, 82*, 85*, 86*, 95*, 99*, 100*.
Homework 10 - in class, included for the Final.
Problem Set 6: 102-105, 109, 111, 112, 116, 118, 119.
Final - Monday, May 2 in MM 207 (same classroom) - Time: 11:00 -1:30