# Math 347 Introduction to proof writing - Fall 2006

Class: MWF 12:00-12:50 in Altgeld Hall 159

and MWF 3:00-3:50 in the Henry Administration Building room 154

Office hours: Mon 10:00-11:00, Tue 15:00-16:00 and Wed 20:00 -21:00 in the Espresso Royale in the Illini Union

The textbook is Mathematical Thinking: Problem-Solving and Proofs by John P.D'Angelo and Douglas B. West.

Homework policy: You are not allowed any late homework. You will however be allowed to drop your lowest scoring homework for the computation of the final grade.

## Exams and homeworks

 What? Due date Comments Percentage median mean standard derivation Homework weekly, due Friday Solutions for last year's Midterm 1 and problem set 5 are now accessible on last year's web-page. Some more solutions are below. 40% First midterm (take-home part) Friday, October 13th Midterm 1 and Solutions 13% First midterm (in-class part) Friday, October 13th Solutions, Last year's Midterm 3, Last year's inclass final. Read Chapter 1-3, the solutions of the problem sets. 7% Second midterm (take-home part) Friday, November 17th Midterm 2 13% Second midterm, in-class part Friday, November 10th 7% Final, take-home part Final 13% Final, in-class part Dec 12th or 15th, 7-10pm 7%

In order to check your scores on-line, go here.

Each of the exams will consist of a take-home and an in-class part. The final will cover the material of both midterms. If you do better on the final part for midterm n than you did in midterm n itself, you may replace the midterm n score by the score in the part of the final corresponding to midterm n. Letter grades will only be computed at the very end of the term, and as follows:
 A A- B+ B B- C+ C D F 85-100 80-84 75-79 70-74 65-69 60-64 55-59 50-54 0-49

## Schedule

 Lecture Date Summary Assignments Suggested reading 1 08/23 Administrative issues. Group work on selected problems. (Here the solutions.) Prepare to present your group's progress to the class. 2 08/25 Presentation and discussion of group work problems. Write a formal proof for Problem 1. 3 08/28 Elementary set theory The chapter about sets 4 08/30 Language and proofs - truth tables The chapter about language and proofs 5 09/01 Language and proofs - quantifiers and negating statements Chapter 2 Problem Set 2 solutions 6 09/06 Language and proofs, negating statements 7 09/08 More on negations; relations and maps Chapter 1, Wikipedia on de Morgan's Laws and propositional logic Problem Set 3 due Friday Sept 15 solutions.) 8 09/11 Injective and surjective maps, students questions on Problem Sets 2 and 3. Book, p.83 9 09/13 Continuation of Monday's discussion, induction. 10 09/15 Induction. 11 09/18 Group work on induction. 12 09/20 Bijections. There is a bijection between two finite sets if and only if the two sets have the same number of elements. Cardinality. 13 09/22 Bijections from the natural numbers to the even numbers, to the odd numbers. Proof that X is countable if and only if X is a union of an increasing sequence of finite sets. A bijection from N x N to N. 14 09/25 Epsilon-delta proofs - how to systematically organize a straightforward proof. Problem Set 4 solutions more solutions 15 09/27 Injective maps have left-inverses. How to systematically organize a straightforward proof (continued). 16 09/29 Injective maps have left-inverses (continued). Questions about Problem Set 4. Problem Set 5 17 10/02 Induction: group work on the lions problem All of Chapter 3 18 10/04 The lions continued. 19 10/06 Review, the handshake problem Book, 3.26. Midterm 1 22 10/13 In-class test Problem Set 6 23 10/16 review of set theory 24 10/18 review set theory 25 10/20 review 26 10/23 modular arithmetic 27 10/25 modular arithmetic 28 10/27 modular arithmetic Problem Set 7 29 10/29 monsters 30 11/1 monsters, finding inverses in modular arithmetic 31 11/3 Groups, statement of Fermat's little theorem Problem Set 8 32 11/5 Groups 33 11/7 Groups, review on set-theory and logic 34 11/10 in-class exam If you want to read more about propositional logic, try the first chapter of Kenneth H. Rosen's book on Discrete Mathematics (it is on reserve in the math library). 35 11/13 groups, subgroups, cosets 36 11/15 The order of a subgroup divides the order of the group 37 11/17 order of subgroup divides order of group (ctd). Fermat's little theorem. 38 11/27 Homework from the book: Exercise A.7.; prove the statement in exercise 13.11. a); prove the statement in exercise 14.8. a)

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