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 2  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 Relations
  • Digraph
  • Digraph Representation of Binary Relation
  • Properties of Binary Relation
  • Equivalence relation
  • Notes 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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