## Math 240 Spring, '12 Introduction to Mathematical Thought  Assignments

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TEXTS: [SOL] The Keys to Advanced Mathematics : Recurrent Themes in Abstract Reasoning by Daniel Solow ( Paperback, BOOKMASTERS,1995 )ISBN:9780964451902
[FET] Proof in Geometry by  A. I. Fetisov (Dover).ISBN:9780486453545
[HOU] How to Think Like a Mathematician by Kevin Houston (Cambridge University Press, 2009) ISBN:9780521719780
[SOS] Set Theory & Related Topics by Seymour Lipschutz  (McGraw-Hill,1998) ISBN:9780070381599

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 day after the assigned day.

Week (Topics and readings)
Tuesday Thursday
1
Introduction/ Reading Math /Start Sets
SOL:1.1
HOU: Ch. 2
Polya: Summary on Problem Solving
1-17
Topic: Introduction and  general remarks.
1-19 Continue work on Class Problem #1 (Moodle)
Start work on PS#1-Problems: SOL 1.1,3,4,5
2. HOU:Ch.1 and 3
SOL:1.2,1.3; 3.1-3.1.2
SOL: 1.4,1.5
Polya: Notation
Polya: Definition

Another Polya Summary
Set Operations

1-24
Topic: Sets and set operations.
Topic: Sets and set inclusion. Begin conditional statements.
PS#1-Problems: SOL 1.1,3,4,5
1-26
Topic: More on sets.What is a proof?
Do
: Proof w/o Words #1.
Do: PS#2.SOL:1.7,1.9-1.14
3. HOU: Ch. 4 and 5
SOL:1.6-1.6.2; 3.1.1- 3.1.4; 1.6.4 ; Problem1.27 sol'n
1-31 Conditional Statements and Truth  Connected to Set Definitions of Union and Intersection.
2-2 Truth Tables,and Universal Quantifiers
Due: PS #3.
SOL:1.15,1.17,1.18,1.21;3.1-3.4
Proof Evaluation #1
4. HOU: Ch 6 and 7 [Note: Be ware of TRUTH TABLES!]
SOL 3.1 [Cont'd., Esp'ly 3.1.4(Cartesian Product) ],1.6.3, 1.6.4
2-7 Conditional, Existential, and Universal Statements. Forward and Backwards. [Starting and Finishing]
The importance of definitions.
2-9 Due:PS #4. SOL: 1.25, 1.28,1.35
:Proof w/o Words #2
5. HOU: Ch.8, 10, 12, 14, 15
SOL:1.6.7; SOL 3.1 [Cont'd., Esp'ly 3.1.4(Cartesian Product) ]
Properties of Set Operation (PSO)
2-14  Proofs about sets. Applications of definitions and direct arguments for conditional statements and universal quantifiers. Definition: Cartesian product of Sets. 2-16 Due: PS #5 SOL: 1.29, 1.30, 1.32;  3.9, 3.11, 3.12 plus
(PSO): Write proofs in English (no logic symbols-only set theory notation) for #7
6. HOU: Ch 16, 17, 18
SOL: 3.1.4, 1.6.7 (again);
SOS:1.1-1.7 Problem 1.12
2-21 Definitions and Proof examples- sets, integers, rational numbers.
Due: PS#6 SOL: 3.7, 3.8
2-23
Proof Evaluation #2
Below this line is not yet assigned!
SOL: 1.6.8, 1.6.9 2-21 Read: DS: 1.6.8, 1.6.9, 1.6.10
PS #8 DS: 1.36, 1.37, 1.43

9-27  Read:  DS: 3.2.1
Polya:Working Backwards ; Reductio... [on Moodle]
Proof w/o Words #3.
PS #9 : 3.13,3.17 (a,b)
7
2-28
Quiz #1 on -line Moodle
10-4 :3.2.3

Polya: Problems to find...prove  [on Moodle]
PS #10:DS:1.45-1.48
Proof Evaluation #3
8 Exam #1: TBA Covers work through TBA
3-7  Read:3.1.3; 3.2.3, pp 161-166
Quiz # 2 on-line Moodle

Proof w/o Words #4.
Spring Break
3-14
9
3-21
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)
Proof Evaluation #4

10
3-28
PS #14. DS:2.7(a,b),2.8,(a,b)2.9,2.10
3-30
Read DS:2.3.1 plus  pp 117-123.
Quiz #3 on-line Moodle (by Monday!)
Proof w/o Words #5.
11
4-3
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

4-5
Notes on Equivalence Relation Example(.pdf).
DS: 1.6.11 plus  pp 117-123. (again)
Continue readings on equivalence relations.
Proof Evaluation #5
12
4-10 Countable and uncountable sets.
Cardinality Reading (on line)
DistributePartnership assignment
4-12
The Real Numbers: Uncountable and countably infinite sets.
Onto Functions and cardinal equivalence.
13 Exam #2 Self-scheduled
Wednesday
4-17
Partnership assignment due by 5 pm.
Basic counting for Finite Sets.
Applications of Counting:
Permutations, Combinations, Counting the Power Sets, Binomial Theorem
4-19
Proof w/o Words #6
The Fundamental Counting Principle
Permutations
Combinations

A proof of the binomial theorem
DS: Read 1.5.1 pp26-28
Optional:Introduction to Trees
The  Division algorithm
Proof in Geometry by Fetisov pp 7-44.

15Final Part I distributed  4-24
DS: pages121-123, 5.1.3
Integer Congruence
Arithmetic and congruence
PS#16-
Rings- Zn, and ring homomorphisms: pi: Z -> Zn.
Start Induction
4-26
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
5-1
5-3

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

DS:1.6.10, 1.6.12  .
DS: 1.43,.1.44, 1.50

Polya:Signs of progress
DS:6.11

More on congruence classes

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

10-25

Do:

Proof Evaluation #7

Do:DS:5.1

DS:p311-312(Symmetry Groups)
handout on Pigeons&Counting
DO: handout:10.1,10.2
Handout on
Do:DS:5.15, 5.16