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.

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Week 1Introduction to Proof I
Week 2Introduction to Proof II
Euclid's lemma
Week 3Introduction to Set Theory I
De Morgan's laws
Week 4Introduction to Set Theory II: Functions and cardinality
Countability of \(\mathbb{Q}\)
First isomorphism theorem of sets
Hilbert's grand hotel
Week 5Proof by Contradiction
Irrationality of \(\sqrt{2}\)
Cantor's diagonal argument
Pigeonhole principle
Euclid's theorem on primes
Week 6Proof by Induction
Well-ordering theorem
Fibonacci sequence
Pick's theorem
Week 7Proofs of Existence and Uniqueness
Fundamental theorem of arithmetic
Koenig's lemma
Division algorithm
Week 8Introduction to Fields
Fermat's little theorem
Freshman's dream
Week 9Morphisms of Fields I
Isomorphisms
Prime fields
Week 10Morphisms of Fields II
Zorn's lemma
Existence of algebraic closures
Week 11Advanced Calculus I: Continuous functions
Week 12Advanced Calculus II: Compact intervals
Intermediate value theorem
Extreme value theorem
Heine-Cantor theorem
Week 13Advanced Calculus III: Differentiation
Differentiation rules
Mean value theorem
l'Hospital's rule
Weeks 14-15Special 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.