the cedar ledge

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

# Bibliography

Not Available

### Motivation

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 â€“ but at the same time exciting. So, this post aims to allay these fears once and for all!

### Project Goals

This process is adapted from the Ultralearning framework posited by Scott Young.

#### What Am I Doing?

Gain an undergraduate level of understanding of proofs and proof creation on par with maths and computer science students.

##### Concepts
• 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 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

• 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 youâ€™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)

DoneTopicOutcomes
XSet TheoryUnderstand basic set notation and operations, including subsets, set identities, and Cartesian products.
XLogicUnderstand propositional logic, including truth tables, implications, negations, and quantifiers.
Proof WritingDevelop skills in writing and constructing mathematical proofs, including direct and indirect proofs, proof by contradiction, proof by contrapositive, and proof by mathematical induction.
Number TheoryUnderstand basic number theory concepts, including divisibility, modular arithmetic, prime numbers, the fundamental theorem of arithmetic, the Euclidean algorithm, and Diophantine equations.
CombinatoricsDevelop skills in combinatorial counting principles, including permutations, combinations, and the inclusion-exclusion principle.
AnalysisDevelop skills in calculus, including limits, continuity, differentiation, integration, sequences, and series.
AlgebraUnderstand 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.

DoneSkill LevelTopicConcept
XBeginnerSet TheorySet Identities
XBeginnerSet TheorySet Notation
XBeginnerSet TheorySet Operations
XBeginnerSet TheorySets and Elements
XBeginnerSet TheorySubsets and Proper Subsets
BeginnerSet TheoryCartesian Products
XBeginnerLogicPropositions and Logical Connectives
XBeginnerLogicTruth Tables
XBeginnerLogicImplication and Equivalence
XBeginnerLogicNegation and De Morgan's Laws
XBeginnerLogicQuantifiers
XBeginnerLogicLogical Implication
XIntermediateProof WritingProof Techniques
IntermediateProof WritingDirect Proofs
IntermediateProof WritingIndirect Proofs
IntermediateProof WritingProof by Contrapositive
IntermediateProof WritingProof by Mathematical Induction
IntermediateProof WritingStrong Induction
IntermediateProof WritingStructural Induction
IntermediateProof WritingProof by Cases
IntermediateProof WritingExistence and Uniqueness Proofs
IntermediateProof WritingCounterexamples
IntermediateNumber TheoryDivisibility and Modular Arithmetic
IntermediateNumber TheoryGCD and LCM
IntermediateNumber TheoryPrime Numbers
IntermediateNumber TheoryFundamental Theorem of Arithmetic
IntermediateNumber TheoryEuclidean Algorithm
IntermediateNumber TheoryDiophantine Equations
IntermediateCombinatoricsCounting Principles
IntermediateCombinatoricsPigeonhole Principle
IntermediateCombinatoricsPermutations and Combinations
IntermediateCombinatoricsInclusion-Exclusion Principle
IntermediateCombinatoricsRecurrence Relations
IntermediateCombinatoricsGenerating Functions

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!

## How To Cite

Zelko, Jacob. Learning What Every Undergraduate Mathematician Should Know about Proofs. https://jacobzelko.com/04012023221538-learning-proofs-beginners. April 1 2023.

## Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: May 19, 2024. Website built with Franklin.jl and the Julia programming language.