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
(McGrawHill,1998) ISBN:9780070381599
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 Representation of Set Equality, Subset, Etc 
821 Topic: Introduction and general remarks. 
823 Continue work on Class Problem #1
(Moodle) Optional online Exercises 1 Click here Start work on PS#1Problems: SOL 1.1,3,4,5 

2. HOU:Ch.1 and 3 SOL:1.2,1.3; 3.13.1.2 SOL: 1.4,1.5 Polya: Notation Polya: Definition Another Polya Summary Set Operations Optional online Exercises 2 Click here Properties of Set Operation Optional Exercises 3 Click here 
828 $$PS#1Problems: SOL 1.1,3,4,5$$ 
830 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:1.61.6.2; 3.1.1 3.1.4; 1.6.4 ; Problem:1.27 sol'n 
94 $$Do: PS#2.SOL:1.7,1.91.14
$$ Topic: More on sets.What is a proof? Conditional Statements and Truth 
96 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 
911 Conditional, Existential, and
Universal Statements. Forward and Backwards. [Starting
and Finishing] The importance of definitions. Do: $$PS #3. SOL:1.15,1.17,1.18,1.21; 3.13.4 $$ 
913 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) 
918 Proofs about
sets. Applications of definitions and direct arguments for
conditional statements and universal quantifiers. 
920Definition:
Cartesian product of Sets. Do: $$PS #5 SOL: 1.29, 1.30, 1.32; 3.9, 3.11, 3.12$$ plus $$(PSO): Write a proof in English (no logic symbolsonly set theory notation) for #7 $$ 

6. HOU: Ch 16, 17, 18 SOL: 3.1.4, 1.6.7 (again); SOS:1.11.7 Problem 1.12 
925 Definitions and
Proof examples sets, integers, rational numbers. $$Do: PS#6 SOL: 3.7, 3.8$$ 
927 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 121(pp. 128) [review with focus on geometry] Polya:Working Backwards ; Reductio... [on Moodle] 
102 Contrapositive. Reductio... Finite vs. infinite sets. Rational vs irrational real numbers. real vs nonreal complex numbers. Empty vs nonempty sets. Start Indirect Arguments. $$DO: PS #7 SOL: 1.36, 1.37 $$ 
104 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 161166, 1.6.10 Polya: Problems to find...prove [on Moodle] Much about functions. 
109 Functions,
Operations, and proofs! $$PS #9: SOL:1.43 1.47, 3.13, 3.17 (b,d)$$ 
38 $$Proof
Evaluation
#3$$ 

9.Exam #1: self scheduled:
Wed. 1017 Covers work through 38 Sign up on MOODLE. HOU: Ch 11, 20, 23, 26 (some review), 30 (again!) Optional:Ch 28 SOL 1.6.12(uniqueness), 3.2.2 plus pp 166171 Optional: 5.1.1 SOS: 4.14.4 Exercises 4.14.3,4.8, 4.18 
1016 Optional :Much about functions Online Exercises (15 only)  1018 $$Proof
w/o Words #4. PS 10 SOL: 3.14, 3.19, 3.25;$$ 

10 HOU: Ch 21, 27, 30 (again!) SOL:2.2.1; 3.2, 5.1.2, 6.24 
1023 FT of Arithmetic. $$ SOL: 2.7(a,b),2.8,(a,b), 2.9, 2.10 $$ 
1025 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 117123. HOU: Ch 31 SOS: 3.3, 3.4, 3.6, 3.8, 3.9 Solved problem: 3.22 Online reading on relations and equivalence relations. 
1030$$ PS#11 [Download .pdf] $$ [Links fixed 10294:00pm]  111 Quiz # 3 online Moodle $$Proof w/o Words #5. PS#12 [Download .pdf](problem 5 due 11/6)$$ Relations 

12 SOL:6.2.4; 1.6.5, pp9496 HOU: p6, pp224227 On a Property of the Class of all Real Algebraic Numbers. by G. Cantor (on Moodle) Pidgeon Hole Principle: I.[cuttheknot] and II [wikipedia] 
116$$
PS
#13 Online Exercises 1,2,5,6 $$ Continue Discussion of Equivalences Relations, Equivalence Classes  start Partitions GCD( r,s) = ar + bs. 
118 Quiz #4 online Moodle on
functions, relations and partitions (by Monday!) $$ PS#14Partitions [Download .pdf]$$ Distribute Partnership assignment 

13 Exam #2
Selfscheduled Wednesday 1114. Sign up on Moodle. HOU Ch 28 esp.pp200303 SOL:5.1.4 The Tower of Hanoi, Cardinality Reading (on line) 
1113 Euclid's Lemma Countable and uncountable sets. The Real Numbers: Uncountable and countably infinite sets. Onto Functions and cardinal equivalence. 
1115Partnership 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! 
1120 
1122 Thanksgiving 

14 Final Part I distributed on Thursday SOL: 1.5.1 pp2628; 1.6.5; 5.1.3 ; 5.3.1 HOU: Ch. 24; pp 224227 On line reading: The Fundamental Counting Principle Permutations Combinations A proof of the binomial theorem 
1127 $$DO
Quiz #4 on MOODLE$$ $$ Proof w/o Words #6 $$ 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. 
1129$$
PS#15Counting
[Download .pdf] Proof Evaluation #5 $$ 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 2844. 
123 
125
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  
Read DS: 5.2.1  Do:DS:5.1  
DS:p311312(Symmetry Groups) handout on Pigeons&Counting 
DO: handout:10.1,10.2  Do:DS:5.15, 5.16 DO: Proof Evaluation #9 Problems on Induction 

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