Math 220 (section 2): Introduction to Mathematical Reasoning, Fall 2021
Professor: Darij Grinberg
Organization
Course description
A transition course that develops the reasoning skills necessary for proof-based courses. Emphasizes writing and presentation skills. Topics taken from set theory, logic, induction, relations, functions, and properties of number sequences and finite sets.
Level: undergraduate.
Prerequisites: MATH 123 [Min Grade: C-] or MATH 200 [Min Grade: C-].
Course materials
- Recommended:
- [GilVan] Will J. Gilbert, Scott A. Vanstone, Introduction to Mathematical Thinking, Pearson 2005: The book I will follow the closest (which doesn't mean much, as I won't really follow any book). Scans of the first 7 chapters; older edition (1993). Errata.
- [KeeGui] Patrick Keef, David Guichard, Introduction to Higher Mathematics, 2021: Too terse to be a textbook, but nice choice of topics.
- [Eccles] Peter J. Eccles, An Introduction to Mathematical Reasoning: numbers, sets and functions, Cambridge University Press 2010: Choice of topics rather similar to ours. Errata.
- [Newste] Clive Newstead, An Infinite Descent into Pure Mathematics, 2021: Goes quite deep and doesn't shy away from harder topics. Unfortunately, still very much a work in progress.
- [DAnWes] John P. D'Angelo, Douglas B. West, Mathematical thinking: problem-solving and proofs, 2nd edition, Prentice Hall 1999: Far more than would fit in a quarter. Errata.
- [Sundst] Ted Sundstrom, Mathematical Reasoning: Writing and Proof, 2021: This is heavy on foundations and takes a long time to get anywhere interesting. On the upside, it appears to be a great place to look up these foundations.
- [Loehr] Nicholas A. Loehr, An Introduction to Mathematical Proofs, CRC Press 2019: Seems similar to [Sundst]. I know Loehr to be a really good writer. Errata.
- [MorMor] David Witte Morris, Joy Morris, Proofs and Concepts: the fundamentals of abstract mathematics, 2017: Another basics- and methods-heavy introductory treatment.
- [Rotman] Joseph J. Rotman, Journey into mathematics: an introduction to proofs, Dover 2007: An approach from the geometric side.
- [LeLeMe] Eric Lehman, F. Thomson Leighton, Albert R. Meyer, Mathematics for Computer Science, 2018: MIT's famous introduction to mathematics for EECS students. While visibly targeting computer scientists, it is a lively introduction to many aspects of mathematics, with high-quality exercises.
- [Aspnes] James Aspnes, Notes on Discrete Mathematics, 2020: Another set of notes written for computer scientists.
- [ForRas] Sylvia Forman, Agnes M. Rash, The Whole Truth About Whole Numbers: An Elementary Introduction to Number Theory, Springer 2015: What the title says. Focusses on elementary number theory (but not on the same parts that we do). Errata.
- [Belcas] Sarah-Marie Belcastro, Discrete mathematics with ducks, 2nd edition, CRC 2019: The first 4 chapters are an introduction to proofs. Errata.
- [Burton] David M. Burton, Elementary number theory, 7th edition, McGraw-Hill 2011: An introduction to elementary number theory that starts rather early and goes rather far. I'll probably use it in some parts of this course.
Course calendar
- Note:
- Week 1:
- Week 2:
- Week 3:
- Week 4:
- Week 5:
- Week 6:
- Week 7:
- Week 8:
- Week 9:
- Plan:
-
The following is a very tentative plan.
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Overview and examples
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The language of logic and sets ([GilVan, Chapter 1])
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Induction ([GilVan, Chapter 4])
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Divisibility and modular congruence ([GilVan, Chapters 2-3])
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Some integer sequences
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Relations ([GilVan, Chapter 3])
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Rational numbers ([GilVan, Chapter 5])
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Finite sets and the pigeonhole principles
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Factorials and binomial coefficients
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Prime numbers
Grading and policies
- Grading matrix:
- 70%: homework.
- 30%: final.
- Grade scale:
- These numbers are tentative and subject to change:
- A+, A, A-: (80%, 100%].
- B+, B, B-: (60%, 80%].
- C+, C, C-: (40%, 60%].
- D+, D, D-: (20%, 40%].
- Homework policy:
- Collaboration and reading is allowed, but you have to write solutions in your own words and acknowledge all sources that you used.
- Asking outsiders (anyone apart from your classmates and Drexel staff) for help with the problems is not allowed. (In particular, you cannot post homework as questions on math.stackexchange before the due date!)
- Late homework will not be accepted.
- Solutions have to be submitted electronically via Gradescope in a format I can read (PDF, TeX or plain text if it works; no doc/docx!). If you submit a PDF, make sure that it is readable and does not go over the margins. If there are problems with submission, send your work to me by email for good measure.
- Expected outcomes:
- The students should have an understanding of mathematical proof and rigor that allows them to read undergraduate-level textbooks, judge the validity of a proof, and write up their own proofs clearly and correctly.
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