MUSA 74: Transition to Upper-Division Mathematics
The transition from lower division to upper division mathematics courses can be quite daunting even to a very experienced student. Unlike other subjects, the difference between lower and upper division courses in mathematics can be quite overwhelming; the two main culprits being writing proofs and abstract concepts. In this course we will address these issues head-on. In particular, we will learn how to write proofs while developing good mathematical style, while teaching students how to work with each other on more difficult problems. We will also give students more familiarity with the mathematical objects appearing in Math 104 and Math 113.
MUSA 74 is a 2-unit DeCal which is intended for students who have no familiarity with writing proofs, and aren't sure if they're prepared enough for upper-division classes. In particular, we strongly recommend that the class is taken alongside Math 53, 54, or 55. We officially assume no prerequisites other than a little calculus (at the level of Math 1A), though we will also appeal to Math 53, 54, and 55 for a few examples. In order to ease the transition, we plan to focus on more of the abstract concepts found in calculus, linear algebra, and differential equations. We will delve into these concepts further by focusing on the proofs that arise when constructing these ideas. By the time you complete this course, you will be comfortable with writing proofs at the level required by the core upper-division sequence of Math 110, Math 113, Math 104, and Math 185.
Fall 2019 Course notes (Spring 2021 coming soon)
Course plan
Here's a rough plan of the course, plus a sample of some of the examples that we'll use.
Week 1 | Introduction to Proof I |
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Week 2 | Introduction to Proof II |
Euclid's lemma |
Week 3 | Introduction to Set Theory I |
De Morgan's laws |
Week 4 | Introduction to Set Theory II: Functions and cardinality |
Countability of \(\mathbb{Q}\) |
First isomorphism theorem of sets |
Hilbert's grand hotel |
Week 5 | Proof by Contradiction |
Irrationality of \(\sqrt{2}\) |
Cantor's diagonal argument |
Pigeonhole principle |
Euclid's theorem on primes |
Week 6 | Proof by Induction |
Well-ordering theorem |
Fibonacci sequence |
Pick's theorem | >
Week 7 | Proofs of Existence and Uniqueness |
Fundamental theorem of arithmetic |
Koenig's lemma |
Division algorithm |
Week 8 | Introduction to Fields |
Fermat's little theorem |
Freshman's dream |
Week 9 | Morphisms of Fields I |
Isomorphisms |
Prime fields |
Week 10 | Morphisms of Fields II |
Zorn's lemma |
Existence of algebraic closures |
Week 11 | Advanced Calculus I: Continuous functions |
Week 12 | Advanced Calculus II: Compact intervals |
Intermediate value theorem |
Extreme value theorem |
Heine-Cantor theorem |
Week 13 | Advanced Calculus III: Differentiation |
Differentiation rules |
Mean value theorem |
l'Hospital's rule |
Weeks 14-15 | Special topics, possibly including: |
Classification of finite fields |
A taste of measure theory |
The Cantor set |
The axiom of choice and its consequences |
Homework
Coming soon.