## Math 240 Fall, '06 Introduction to Mathematical Thought  Assignments

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TEXTS:  The Keys to Advanced Mathematics : Recurrent Themes in Abstract Reasoning by Daniel Solow ( Paperback,  Books Unlimited,1995 )
Proof in Geometry by  A. I. Fetisov (Out of Print- Bookstore Reprint, D. C. Heath, Boston, 1963).

Assignments - (subject to change)
Problems are due on the class day for which they are listed.
All assignments are tentative until assigned a PS#.
Show all work and explain your reasoning
Late homework is not accepted after 5 pm of the due day.
DS = The Keys to Advanced Mathematics; F =Proof in Geometry
Week Monday Wednesday Friday
1 8-21
Topic: Introduction and  general remarks.
8-23 Read :Polya: Summary on Problem Solving
DS:1.1
Introduction to Sets
Representation of Set
Basics(Equality, Subset, Etc)
Optional on-line exercises 1
PS#1.Problems:DS 1.1,3,4,5
8-25 Topic: Sets and set operations.
Set Operations
Optional on-line exercises 2
DS:1.2,1.3; 3.1-3.1.2
Do: PS#2.DS:1.7,1.9-1.14
2.
8-28
Topic: Sets and set inclusion. Begin conditional statements.
Properties of Set Operation

8-30 Topic: More on sets.What is a proof?

Polya: Notation  [new]
Polya: Definition  [new]

DS:1.6-1.6.2
Do
:Proof w/o Words #1.
Do: PS #3. DS:1.15,1.17,1.18,1.21,1.22;3.1-3.4
3 9-4 Labor Day NO CLASS.
9-6 Truth Tables, Conditional statements  and Universal Quantifiers Connected to Set definitions of Inclusion and Equality.
9-8
Do: PS #4. DS: 1.25,1.27, 1.28,1.35
Do Proof Evaluation #1
4
9-11
Read : Properties of Set Operation (PSO)
DS:3.1
DS: 1.6.3, 1.6.4
Do:PS #5 DS: 3.9,3.11, 3.12 plus
(PSO): Write proofs in English (no logic symbols-only set theory notation) for #7 . [Changed sept. 11]
9-15  "The Proof"
Write 4 questions based on "the proof.".
5 9-18
DS: 1.6.3,1.6.4 (again!)
3.1.4(cartesian Product)
DO:Proof w/o Words #2
D.S:1.6.7
Problems:
PS #6 DS: 1.29, 1.30, 1.32
DS: 1.6.8, 1.6.9
PS#7 DS: 3.7 a, 3.8a
Proof Evaluation #2
9-25 Read: DS: 1.6.8, 1.6.9, 1.6.10
PS #8 DS: 1.36, 1.37, 1.43

Resubmit 3.12 and
(PSO) #7 : Write proofs in English (no logic symbols-only set theory notation).
also 1.32
Polya:Working Backwards ; Reductio... [on Moodle]
Proof w/o Words #3.
PS #9 : 3.13,3.17 (a,b)
7
Quiz #1 on -line Moodle
10-4 :3.2.3
10-6

Polya: Problems to find...prove  [on Moodle]
PS #10:DS:1.45-1.48
Proof Evaluation #3
8 Exam #1: Thursday 10-12. Covers work through PS #11.
Quiz # 2 on-line Moodle

10-13
Proof w/o Words #4.
9
10 -16
Read DS: 1.6.12; 3.2.2  plus pp 166-171
PS #12. 3.25, 3.26
10-18
Optional On-line Exercises (1-5 only)

10-20
Proof Evaluation #4
10
10-23
PS #14. DS:2.7(a,b),2.8,(a,b)2.9,2.10
10-25
10-27
Quiz #3 on-line Moodle (by Monday!)
Proof w/o Words #5.
11
10-30
Read DS: 1.5.1; :2.3.1 plus  pp 117-123. (again)
PS#15
-

On-line reading on relations, digraphs, and equivalence relations.

#### Introduction to Relation Binary Relation Definition of Relation (general relation) Equality of RelationsDigraphDigraph Representation of Binary RelationProperties of Binary RelationEquivalence relationOptional On-line Exercises 1,2,5,6

11-1
Notes on Equivalence Relation Example(.pdf).
DS: 1.6.11 plus  pp 117-123. (again)
11-3 Continue readings on equivalence relations.
Proof Evaluation #5

12
11-6 Countable and uncountable sets.
DistributePartnership assignment
11-8 The Real Numbers: Uncountable and countably infinite sets.
Onto Functions and cardinal equivalence.
11-10 No Class  Holiday
13 Exam #2 Self-scheduled
Tuesday
11-13
Partnership assignment due by 5 pm.
Basic counting for Finite Sets.
11-15 Applications of Counting:
Permutations, Combinations, Counting the Power Sets, Binomial Theorem

11-17Proof w/o Words #6
The Fundamental Counting Principle
Permutations
Combinations

A proof of the binomial theorem
Optional:Introduction to Trees
The  Division algorithm

Proof in Geometry by Fetisov pp 7-44.
14 Fall Break
11-2011-2211-24
15Final Part I distributed 11-27 DS: pages121-123, 5.1.3
Integer Congruence
Arithmetic and congruence
Rings- Zn, and ring homomorphisms: pi: Z -> Zn.
Start Induction
DS: 1.6.5; 5.3.1
More on Induction- Well Ordering
The Tower of Hanoi,
Pidgeon Hole Principle: I.[cut-the-knot]  and II [wikipedia]
PS #17 DS:1.33(b); 1.34; 5.2
Proof Evaluation #6
Distribute Final I
16
Last week of classes
12-4
12-6 12-8
Below this line is not yet assigned!

, 1.50
Do:
DS:3.25, 3.26
Problems: DS:3.2.3
DO:

9- 6
DS:1.6.10, 1.6.12  .
DS: 1.43,.1.44, 1.50

Polya:Signs of progress 10-16

DO:

DS:6.11

10 10-23 More on congruence classes

DS:6.2.4 (this should cover several classes)
DO:

10-25

Do:

Proof Evaluation #7

11

11-3
Do:DS:5.1

12

11-10 No Class  Holiday

DO
15Final Part I distributed  11-27 READ
DS:p311-312(Symmetry Groups)
handout on Pigeons&Counting
DO: handout:10.1,10.2
Handout on
Do:DS:5.15, 5.16