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
Introduction to Set Theory  Click here
Representation of Set Click here
Equality, Subset, Etc Click here
1-17
Topic: Introduction and  general remarks.
1-19 Continue work on Class Problem #1 (Moodle)
Optional on-line Exercises 1 Click here
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
Click here
Optional on-line Exercises Click here
Properties of Set Operation Click here
Optional Exercises 3 Click here

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?
Read :
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 Complications with quantifiers.
Proof Evaluation #2

7  Indirect arguments, Functions
HOU:  Reread Ch 1 (esp'lly pp 10,11)
SOL: 1.6.8, 1.6.9 , 1.6.10; 3.2.1,  3.2.2
FET: Articles 1-21(pp. 1-28) [review with focus on geometry]
Polya:Working Backwards ; Reductio... [on Moodle]
2-28 Contrapositive. Reductio...
Finite vs. infinite sets.
Rational vs irrational real numbers.
real vs non-real complex numbers.
Empty vs non-empty sets.

Start Indirect Arguments.
Due: PS #7 SOL: 1.36, 1.37, 1.43
3-1 Contrapositive. Reductio...
When is something "Well defined"? Operations and Functions.
Quiz #1 on -line Moodle
Due: PS #8 SOL : 3.13, 3.17 (b,d)
[Changed 2-28]
Proof w/o Words #3.

8. Exam #1: self scheduled: Wed. 3-7 Covers work through 3-1 Sign up on MOODLE.
Functions!
HOU: Ch 30
SOL:3.1.3, 3.2.3 pp 161-166, 1.6.10
Polya: Problems to find...prove  [on Moodle]
3-6 Functions, Operations, and proofs!
Due: PS #9: SOL:1.43 -1.47
3-8 Proof Evaluation #3
Spring Break: Start work on week 9 / Catch up on previous reading!
3-13 3-15

9.
HOU: Ch 11, 20, 23, 26 (some review), 30 (again!) Optional:Ch 28
SOL  1.6.12(uniqueness), 3.2.2 plus pp 166-171 Optional: 5.1.1
SOS: 4.1-4.4 Exercises 4.1-4.3,4.8, 4.18
Much about functions.
3-20
Due: PS 10 SOL: 3.14, 3.19, 3.25;
[Ignore-assignment error HOU: Exercises 3.8 (ii, v, xii)  3-10-2012]
Optional :Much about functions On-line Exercises (1-5 only)
3-22 Proof w/o Words #4.

10 HOU: Ch  21, 27, 30 (again!)
SOL:2.2.1; 3.2, 5.1.2, 6.24
3-28
PS#11- [Download .pdf] plus SOL: 2.7(a,b),2.8,(a,b), 2.9, 2.10
Quiz # 2 on-line Moodle
Due Wed.
3-30
PS#12- [Download .pdf]
Proof Evaluation #4

11 Polya: Signs of progress (on Moodle)
SOL: 1.5.1; 1.6.11; 2.3.1 plus  pp 117-123.
HOU: Ch 31
SOS: 3.3, 3.4, 3.6, 3.8, 3.9 Solved problem: 3.22
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 relationNotes on Equivalence Relation Example(.pdf).

4-3 PS #13 On-line Exercises 1,2,5,6
4-5 PS#14-Partitions [Download .pdf]
Proof w/o Words #5.

12 SOL:6.2.4; 1.6.5, pp94-96
HOU: p6, pp224-227
On a Property of the Class of all Real Algebraic Numbers. by G. Cantor (on Moodle)
Pidgeon Hole Principle: I.[cut-the-knot]  and II [wikipedia]
4-10 Continue Discussion of Partitions and Relations

Countable and uncountable sets.
Distribute Partnership assignment
4-12 Quiz #3 on-line Moodle on functions, relations and partitions (by Monday!)
The Real Numbers: Uncountable and countably infinite sets.
Onto Functions and cardinal equivalence.

13 Exam #2 Self-scheduled
Wednesday 4-18. Sign up on Moodle.
HOU Ch 28 esp.pp200-303
SOL:5.1.4
The Tower of Hanoi,
Cardinality Reading (on line)
4-17Partnership assignment due by 5 pm.
Uniqueness in the FT of Arithmetic.

Basic counting for Finite Sets.
Applications of Counting:
4-19
Proof w/o Words #6
Counting continued.
Permutations, Combinations
Start Discussion of uncountable infinite sets

14 Final Part I distributed on Thursday
SOL: 1.5.1 pp26-28; 1.6.5; 5.1.3 ; 5.3.1
HOU: Ch. 24; pp 224-227
On line reading:
The Fundamental Counting Principle
Permutations
Combinations

A proof of the binomial theorem
4-24
More Tower of Hanoi: Start Induction as a proof method.
Counting the Power Sets, Binomial Theorem
Integer Congruence
Arithmetic and congruence
Rings- Zn, and ring homomorphisms: pi: Z -> Zn.
4-26 PS#15-Counting [Download .pdf]
More on Induction- Well Ordering
Distribute Final I part I

Below this line is not yet assigned!

FET pp 28-44.

Proof Evaluation #5

15
Last week of classes
5-1
5-3PS #17 SOL:1.33(b); 1.34; 5.2
Proof Evaluation #6

16 Final Examination Self scheduled
Review Session:
Sunday 5-6
TBA

5-8
FOR 107:  1500-1700
5-10
ARTA_027    0800-1000
5-11
FH 177:   1020-1220
FOR 107:  1500-1700

, 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

DS:6.11

More on congruence classes

Read

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

10-25
Read:

Do:

Proof Evaluation #7

Do:DS:5.1

Read DS: 5.2.1

READ
DS:p311-312(Symmetry Groups)
handout on Pigeons&Counting
DO: handout:10.1,10.2
READ
Handout on
Do:DS:5.15, 5.16
Read:
DO: Proof Evaluation  #9
Problems on Induction
Distribute Final I

Read: Handout on graphs, combinations.
DO: 4 induction problems on sheet
Read:

DO:

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