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
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
Read:  DS: 3.2.1


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
 Read 2.1.1,1.6.10(negation)
Quiz #1 on -line Moodle
10-4 :3.2.3
Read: 3.2.3, 1.6.10

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
10-11 Read:1.6.12(uniqueness)
PS#11- [Download .pdf]

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
Read DS: 1.6.12
Much about functions.
Optional On-line Exercises (1-5 only)
PS#13- [Download .pdf]
Proof Evaluation #4

10
3-28
Read: DS:2.2.1 
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
-
Partitions [Download .pdf]

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

  1. Introduction to Relation
  2. Binary Relation
  3. Definition of Relation (general relation)
  4. Equality of Relations
  5. Digraph
  6. Digraph Representation of Binary Relation
  7. Properties of Binary Relation
  8. Equivalence relation
  9. Optional On-line Exercises 1,2,5,6

4-5
Read:
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
On line reading:
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-
Counting [Download .pdf]
 Read DS:5.1.1;5.1.2; 5.1.4
Rings- Zn, and ring homomorphisms: pi: Z -> Zn.
Start Induction
4-26
Read: 
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 

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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