Date: April 1 2023
Summary: Ultralearning project to learn the equivalent of an undergraduate maths or computer science student understanding of proofs and how to write them.
Keywords: #zettel #archive #project #blog #proof
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As I am personally moving into applied mathematics for my future graduate studies, I realized I had a deficiency. And that deficiency? How to prove things! "Proof" was always a somewhat scary thing that I encountered in mathematics every once in a while – but at the same time exciting. So, this post aims to allay these fears once and for all!
This process is adapted from the Ultralearning framework posited by Scott Young.
Gain an undergraduate level of understanding of proofs and proof creation on par with maths and computer science students.
Sets and Set Operations
Set Notation
Union and Intersection
Complement and Subset
Power Sets
Logical Structure in Proofs
Implication and Equivalence
Logical Connectives (And, Or, Not)
De Morgan's Laws
Contrapositive, Converse, and Negation of Conditional Statements
Quantifiers
Universal Quantifier (For All)
Existential Quantifier (There Exists)
Uniqueness Quantifier (There Exists Exactly One)
Proof Techniques
Direct Proof
Proof by Contradiction
Proof by Contrapositive
Existence Proofs
Induction and Recursion
Principle of Mathematical Induction
Strong Induction
Recursive Definitions
Relations and Functions
Cartesian Product
Equivalence Relations
Partial and Total Orders
Injectivity, Surjectivity, and Bijectivity of Functions
Cardinality and Countability
Cardinality of Sets
Countable and Uncountable Sets
Diagonalization Argument
Advanced Topics
Constructive Mathematics
Proof Assistants
Zorn's Lemma and Well-Ordering Principle
Completeness Axiom and Real Numbers
This is based on the Meta Learning step Young described as well as some additional tweaks of my own:
Outcomes: The knowledge and abilities you’ll need to acquire for success.
Topic: The topic to learn
Done: If this task has been completed (X) or not yet (cell is empty)
| Done | Topic | Outcomes |
|---|---|---|
| X | Set Theory | Understand basic set notation and operations, including subsets, set identities, and Cartesian products. |
| X | Logic | Understand propositional logic, including truth tables, implications, negations, and quantifiers. |
| Proof Writing | Develop skills in writing and constructing mathematical proofs, including direct and indirect proofs, proof by contradiction, proof by contrapositive, and proof by mathematical induction. | |
| Number Theory | Understand basic number theory concepts, including divisibility, modular arithmetic, prime numbers, the fundamental theorem of arithmetic, the Euclidean algorithm, and Diophantine equations. | |
| Combinatorics | Develop skills in combinatorial counting principles, including permutations, combinations, and the inclusion-exclusion principle. | |
| Analysis | Develop skills in calculus, including limits, continuity, differentiation, integration, sequences, and series. | |
| Algebra | Understand algebraic structures, including groups, rings, and fields. |
This table describes the very broad topics, resources I'll use, and the expected learning outcomes for each topic. As I progress through this table, I will add an "X" to each row I have studied. Furthermore, the table is ordered by level of difficulty.
| Done | Skill Level | Topic | Concept |
|---|---|---|---|
| X | Beginner | Set Theory | Set Identities |
| X | Beginner | Set Theory | Set Notation |
| X | Beginner | Set Theory | Set Operations |
| X | Beginner | Set Theory | Sets and Elements |
| X | Beginner | Set Theory | Subsets and Proper Subsets |
| Beginner | Set Theory | Cartesian Products | |
| X | Beginner | Logic | Propositions and Logical Connectives |
| X | Beginner | Logic | Truth Tables |
| X | Beginner | Logic | Implication and Equivalence |
| X | Beginner | Logic | Negation and De Morgan's Laws |
| X | Beginner | Logic | Quantifiers |
| X | Beginner | Logic | Logical Implication |
| X | Intermediate | Proof Writing | Proof Techniques |
| Intermediate | Proof Writing | Direct Proofs | |
| Intermediate | Proof Writing | Indirect Proofs | |
| Intermediate | Proof Writing | Proof by Contradiction | |
| Intermediate | Proof Writing | Proof by Contrapositive | |
| Intermediate | Proof Writing | Proof by Mathematical Induction | |
| Intermediate | Proof Writing | Strong Induction | |
| Intermediate | Proof Writing | Structural Induction | |
| Intermediate | Proof Writing | Proof by Cases | |
| Intermediate | Proof Writing | Existence and Uniqueness Proofs | |
| Intermediate | Proof Writing | Counterexamples | |
| Intermediate | Number Theory | Divisibility and Modular Arithmetic | |
| Intermediate | Number Theory | GCD and LCM | |
| Intermediate | Number Theory | Prime Numbers | |
| Intermediate | Number Theory | Fundamental Theorem of Arithmetic | |
| Intermediate | Number Theory | Euclidean Algorithm | |
| Intermediate | Number Theory | Diophantine Equations | |
| Intermediate | Combinatorics | Counting Principles | |
| Intermediate | Combinatorics | Pigeonhole Principle | |
| Intermediate | Combinatorics | Permutations and Combinations | |
| Intermediate | Combinatorics | Inclusion-Exclusion Principle | |
| Intermediate | Combinatorics | Recurrence Relations | |
| Intermediate | Combinatorics | Generating Functions | |
| Advanced | Analysis | Limits and Continuity | |
| Advanced | Analysis | Differentiation | |
| Advanced | Analysis | Integration | |
| Advanced | Analysis | Sequences and Series | |
| Advanced | Algebra | Groups | |
| Advanced | Algebra | Rings and Fields |
This table gets more into exact topics and concepts to master. They have an associated difficult level and overall topic. Moreover, this a synthesis of concepts and topics to be covered based on class syllabi from:
MA307: Introduction to Proof (taught by Dan Dugger at University of Oregon)
300:T6 Introduction to Mathematical Reasoning (taught by Chloe Urbanski Wawrzyniak at Rutgers University)
Introduction to Proof-based Discrete Mathematics (taught by Matthew Gelvin at University of Chicago)
MATH 301: Introduction To Proofs (taught by Emily Riehl at John Hopkins University)
Additionally, in the construction of this project, I'd like to thank John Carlos Baez for some of his suggestions!
Zelko, Jacob. Learning What Every Undergraduate Mathematician Should Know about Proofs. https://jacobzelko.com/04012023221538-learning-proofs-beginners. April 1 2023.