Math 308, Fall 2018

Bridge to Advanced Mathematics

### Logistics

**Professor:**Zajj Daugherty

**Class:**TuTh 9:30am–10:45am in NAC 6/113

**Office hours:**Thursday 11am-12pm in NAC 6/301. By arrangement only: Tuesday 4pm.

**Textbooks:**How to Think Like a Mathematician: A Companion to Undergraduate Mathematics, by Kevin Houston; Introduction to Mathematical Structures and Proofs, by Larry J. Gerstein; and Elementary Analysis: The Theory of Calculus, by Kenneth A. Ross.

**Grades:**Your grades will be based on homework assignments (15%), a portfolio (20%), and exams (65%). Exams: October 16 & 18, and December 6 & 11.

**Homework 0:** Due Tuesday 9/4. Email zdaugherty@gmail.com from your preferred email account with subject line "Math 308: Homework 0", including the following information:

(a) What name you like to go by, and how is it pronounced.

(b) What you're majoring in, and why.

(c) What your general long-term goals are.

(d) Something that you're really good at.

(e) A photo or a physical description of yourself to help me learn your name faster. ("That guy who doesn't say anything" isn't sufficient.)

### Schedule

Week 1 (8/28&30) |
Read Preface and Chapters 1 & 2 of "How to think..." Notes: [Tuesday (slides) (printout)] [Thursday (slides) (printout)] Articles from class (access from on campus): A short elementary proof, Ramanujan, Rational Ratios, Lebesgue's Road, A Continuous Function. |

Due 9/4: Homework 1 (pdf) (tex) (zip). Solutions: (pdf) (tex) (zip).
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Due 9/13: Homework 2 (pdf) (tex) (zip)
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Week 2 (9/4&6) |
Read Chapters 3 and 4 of "How to think...", and Guidelines for Good Mathematical Writing for Tuesday; Chapter 5 for Thursday. Notes: [Tuesday (slides) (printout)] [Thursday (slides) (printout)] Proofs without words: Problem 1 (image), Problem 2 (image), Problem 3 (image). |

Due 9/20: Homework 3 (pdf) (tex) (zip)Solutions: (pdf) (tex) (zip) | |

Week 3 (9/13) |
No class on TuesdayRead Chapters 6 and 7 of "How to think...". See also Section 1.1–1.3 of "Intro to Math Structures" Notes: [Thursday (slides) (printout)] |

Due 9/27: Homework 4 (pdf) (tex) (zip).
Solutions: (pdf) (tex) (zip)
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Week 4 (9/20) |
No class on TuesdayRead Chapters 8 and 9 of "How to think...". Notes: [Thursday (slides) (printout)] |

Due 10/2: Homework 5 (pdf) (tex) (zip) [Add-ons for Tuesday: (pdf) (tex) (zip)] Solutions: (pdf) (tex) (zip)
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Week 5 (9/25&27) |
Read Chapters 10 and 11 of "How to think...", and Proofs portfolio—initial guidelines. See also Section 2.3 of "Intro to Math Structures" Notes: [Tuesday (slides) (printout)] Thursday: Bring any progress of Homework 5, paper or a computer on which to work, and either a copy of or access to the proofs portfolio handout. |

Week 6 (10/2&4) |
Read Chapters 20 & 21 of "How to think...". See also Section 2.3 of "Intro to Math Structures" Patterns that eventually fail, The Pattern Does Not Hold Notes: [Tuesday (slides) (notes)] [Thursday: Proof Lab I—Direct proofs (pdf) (tex) (zip)] |

Due 10/9: Homework 6 (pdf) (tex) (zip). Solutions: (pdf) (tex) (zip) | |

Week 7 (10/9&11) |
Read Chapters 12—16 of "How to think...". See also Section 2.3 of "Intro to Math Structures" Notes: [Thursday (slides) (notes)] |

Due 10/16: 8 stars worth of proofs from Proof Lab I, including first drafts with peer edits. (Hints)
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Exam 1: 10/16 and 10/18. Covering chapters 1—13 and 20—21. Topics include sets (definitions, set operations), logic (statements, operations, implications and manipulating implications, truth tables, quantifiers), solving problems, writing proofs, reading mathematics (particularly definitions and theorems). Solutions will be written by hand, and LaTeX skills will not be tested. (Exam 1, part 2 -- prompts and hints)
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Week 9 (10/23&25) |
Read Chapters 17—19 and 22 of "How to think...". Notes: [Tuesday (slides) (notes) (worksheet) (Chapter 19)] [Thursday (slides) (notes)] |

Due 10/30: Homework 7 UPDATED (pdf) (tex) (zip).
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Week 10 (10/30&11/1) |
Read Chapter 24 of "How to think..."; see also Section 2.10 in "Intro to Mathematical Structures...". [Tuesday (slides) (notes)] [Thursday: Proof Lab II—Induction (pdf) (tex) (zip) (Hints)] |

Due 11/6: Homework 8 (pdf) (tex) (zip). Solutions: (pdf) (tex) (zip) | |

Week 11 (11/5&11/7) |
Read Chapters 25, 27, and 28 of "How to think..." (skip infinite number of primes [for now] and Diophantine equations); see also Section 2.10 and Chapter 6 of "Intro to Mathematical Structures...". [Tuesday (slides) (notes)] [Tuesday (slides) (notes)] |

Due 11/15(Th): Final draft of 8 stars worth of Proof Lab II—Induction. (Hints)
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Due 11/20 (Tu): Homework 9 (pdf) (tex) (zip).
Solutions: (pdf) (tex) (zip) | |

Week 12 (11/13&11/15) |
Read Chapters 23 and 26 of "How to think...". Also, proof portfolio instructions (final). [Tuesday (slides) (notes)] [Thursday: Proof Lab III—Contradiction (pdf) (tex) (zip) (Hints)] |

Week 13 (11/20) |
Read Chapter 29 of "How to think..."; see also Section 6.4 of "Intro to Mathematical Structures...". [Tuesday (slides) (notes)] |

Due 11/29(Th): Final draft of 8 stars worth of Proof Lab III—Contradiction. (Hints)
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Due 12/4 (Tu): Homework 10 FINAL VERSION (pdf) (tex) (zip)
Solutions: (pdf) (tex) (zip) | |

Week 14 (11/27 & 29) |
Read Chapter 30 & 31 of "How to think...", Chapters 1–4 of "Elementary Analysis". [Tuesday (slides) (notes)] [Thursday (slides) (notes)] |

Week 15 (12/4 & 6) |
Read Chapters 1–4 and 6 of "Elementary Analysis". [Tuesday (slides) (notes)] |

Due 12/11 (Tu): Homework 11 (pdf) (tex) (zip). | |

Exam 2: 12/6 and 12/11. Covering chapters 1, 12, 14—31 in "How to think..." and 1—4 and 6 of "Elementary analysis". Also, a working knowledge of the rest of "How to think" (i.e. you won't be tested on truth tables, but you should have a working knowledge of logical statements and equivalencies). You will be given statements to prove on Thursday 12/6, which may inlude proof techniqes such as proof by direct proof, contrapositive, cases, induction (not strong induction), contradiction. Short-answer and computational questions will be given on Tuesday 12/11; topics may include sets, how to read proofs/theorems/definitions, elementary number theory (divisors, Euclidean algorithm, modular arithmetic), functions and sizes of sets (types of functions, countable/uncountable), relations (binary relations, equivalence relations, equivalence classes), number systems (natural numbers, integers, rational numbers, real numbers, orperations and comparisons). Solutions will be written by hand, and LaTeX skills will not be tested.
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(Some comic relief from studying proof by contradiction) | |

Due 12/18, noon: Proofs portfolio. You can put it under my office door, or email it to me. Template: (tex) (zip). Instructions: (pdf). See "commented out" text (stuff with %'s before it, so that it doesn't print when you compile) in the template for more instructions on how to fill out the template. |

### LaTeX guide

You will be expected to learn to type up your homework using LaTeX. You can either downlad and install LaTeX on your computer, or use a free online compiler like overleaf or share LaTeX (lots of other suggestions).To get you started, download Homework 1 and try to get it to complile. There are also lots of great resources out there to help you learn. To see my code, the LaTeX files for notes and handouts on any of my other course pages can be found by replacing .pdf with .tex for most of my files. You can find another sample on my teaching page, and lots of sample code at TeXample.net. The Not So Short guide to LaTeX is linked from my resources page. You can also look up symbols by drawing at detexify.

More guides to LaTeX, what it is, and how to get started:

- LaTeX beginners tutorial

- Introduction to LaTeX (Short Course given at the UIUC REU Number Theory Program, Summer 2001 - revised August 2004)

- Getting Started with LaTeX, by David R. Wilkins

- Overleaf's "getting started" help section

- YouTube videos from John Mayberry or Michelle Krummel