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

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 marked with "\$\$"
Show all work and explain your reasoning
Late homework is not accepted after 5 pm of the day after the assigned day.

Tuesday Thursday
1
SOL:1.1
HOU: Ch. 2
Polya: Summary on Problem Solving
8-21
Topic: Introduction and  general remarks.
8-23 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

8-28

\$\$PS#1-Problems: SOL 1.1,3,4,5\$\$
8-30
Unification and generalization.
Topic: Sets and set operations.
Topic: Sets and set inclusion.
Begin conditional statements.
\$\$ Do: Proof w/o Words #1. \$\$

3. HOU: Ch. 4 and 5
SOL:1.6-1.6.2; 3.1.1- 3.1.4; 1.6.4 ;
Problem:1.27 sol'n
9-4 \$\$Do: PS#2.SOL:1.7,1.9-1.14 \$\$
Topic: More on sets.What is a proof?  Conditional Statements and Truth
9-6 Truth Tables,and Universal Quantifiers
Connected to Set Definitions of Union and Intersection.
\$\$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
9-11 Conditional, Existential, and Universal Statements. Forward and Backwards. [Starting and Finishing]
The importance of definitions.

Do: \$\$PS #3. SOL:1.15,1.17,1.18,1.21; 3.1-3.4 \$\$
9-13 More on understanding statements: existential and universal.
Do:
\$\$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)
9-18  Proofs about sets. Applications of definitions and direct arguments for conditional statements and universal quantifiers.
9-20Definition: Cartesian product of Sets.
Do: \$\$PS #5 SOL: 1.29, 1.30, 1.32;  3.9, 3.11, 3.12\$\$ plus
\$\$(PSO): Write a proof 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
9-25 Definitions and Proof examples- sets, integers, rational numbers.
\$\$Do: PS#6 SOL: 3.7, 3.8\$\$
9-27 Complications with quantifiers.
\$\$ Do: Proof Evaluation #2\$\$
Quiz #1 on -line Moodle By Friday

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]
10-2
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.
\$\$DO: PS #7 SOL: 1.36, 1.37 \$\$
10-4 Contrapositive. Reductio...
When is something "Well defined"?
Operations and Functions.

\$\$Do: PS #8 SOL :1.43
Resubmit 3.12
, PSO #7
Proof w/o Words #3.\$\$

8. Functions!
HOU: Ch 30
SOL:3.1.3, 3.2.3 pp 161-166, 1.6.10
Polya: Problems to find...prove  [on Moodle]
10-9 Functions, Operations, and proofs!
\$\$PS #9: SOL:1.43 -1.47, 3.13, 3.17 (b,d)\$\$
3-8 \$\$Proof Evaluation #3\$\$

9.Exam #1: self scheduled: Wed. 10-17 Covers work through 3-8
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
10-16  Optional :Much about functions On-line Exercises (1-5 only) 10-18  \$\$Proof w/o Words #4.
PS 10 SOL: 3.14, 3.19, 3.25;\$\$

10 HOU: Ch  21, 27, 30 (again!)
SOL:2.2.1; 3.2, 5.1.2, 6.24
10-23
FT of Arithmetic.
\$\$ SOL: 2.7(a,b),2.8,(a,b), 2.9, 2.10 \$\$
10-25  The division algorithm.
The set or primes is infinite.
Composition of functions, bijections and inverse functions.
\$\$ 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 and equivalence relations.

#### Introduction to Relation Binary Relation Definition of Relation (general relation) Equality of RelationsDigraphDigraph Representation of Binary RelationProperties of Binary RelationEquivalence relation

\$\$Proof w/o Words #5.
(problem 5 due 11/6)\$\$
Relations

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]
11-6\$\$  PS #13 On-line Exercises 1,2,5,6 \$\$
Continue Discussion of Equivalences Relations, Equivalence Classes - start Partitions
GCD( r,s) = ar + bs.

11-8 Quiz #4 on-line Moodle on functions, relations and partitions (by Monday!)

Distribute Partnership assignment

13 Exam #2 Self-scheduled
HOU Ch 28 esp.pp200-303
SOL:5.1.4
The Tower of Hanoi,
11-13 Euclid's Lemma
Countable and uncountable sets.
The Real Numbers: Uncountable and countably infinite sets.
Onto Functions and cardinal equivalence.
11-15Partnership assignment due by 5 pm Uniqueness in the FT of Arithmetic.
Counting continued.
Uncountable infinite sets
The Real Numbers: Uncountable and countably infinite sets.
Unions and intersections for ["large"] families of sets.

Break: Start work on week 14 / Catch up on previous reading!

11-20
11-22 Thanksgiving

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
The Fundamental Counting Principle
Permutations
Combinations

A proof of the binomial theorem
11-27 \$\$DO Quiz #4 on MOODLE\$\$
\$\$ Proof w/o Words #6 \$\$
Basic counting for Finite Sets.
Begin Applications of Counting:
Permutations, Combinations
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.
Proof Evaluation #5 \$\$
More on Induction- Well Ordering
Distribute Final I part I

Below this line is not yet assigned! [Subject to change.]
15
Last week of classes
FET pp 28-44.
12-3
12-5 PS #16 SOL:1.33(b); 1.34; 5.2
Proof Evaluation #6

16 Final Examination Self scheduled
Review Session:
Sunday
TBA

Possible Further Work

DS: 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

DS:6.2.4 (this should cover several classes)
Proof Evaluation #7