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
| 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 |
8-21 Topic: Introduction and general remarks. |
8-23 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 Optional on-line Exercises 2 Click here Properties of Set Operation Optional Exercises 3 Click here |
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] Much about functions. |
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 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 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. |
10-30$$ PS#11- [Download .pdf] $$ [Links fixed 10-29-4:00pm] | 11-1 Quiz # 3 on-line Moodle $$Proof w/o Words #5. PS#12- [Download .pdf](problem 5 due 11/6)$$ 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!) $$ PS#14-Partitions [Download .pdf]$$ Distribute Partnership assignment |
|
| 13 Exam #2
Self-scheduled Wednesday 11-14. Sign up on Moodle. HOU Ch 28 esp.pp200-303 SOL:5.1.4 The Tower of Hanoi, Cardinality Reading (on line) |
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 On line reading: 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. |
11-29$$
PS#15-Counting
[Download .pdf] Proof Evaluation #5 $$ 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 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 |
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| Possible Further Work |
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| 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:p311-312(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: |
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