the cedar ledge

Learning What Every Undergraduate Mathematician Should Know about Proofs

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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Table of Contents

    1. Motivation
    2. Project Goals
      1. What Am I Doing?
        1. Concepts
    3. Roadmap
  1. How To Cite
  2. References:
  3. Discussion:


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!

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.



This is based on the Meta Learning step Young described as well as some additional tweaks of my own:

Set TheoryUnderstand basic set notation and operations, including subsets, set identities, and Cartesian products.
LogicUnderstand 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
BeginnerSet TheorySets and Elements
BeginnerSet TheorySet Notation
BeginnerSet TheorySubsets and Proper Subsets
BeginnerSet TheorySet Operations
BeginnerSet TheorySet Identities
BeginnerSet TheoryCartesian Products
BeginnerLogicPropositions and Logical Connectives
BeginnerLogicTruth Tables
BeginnerLogicImplication and Equivalence
BeginnerLogicNegation and De Morgan's Laws
BeginnerLogicLogical Implication
IntermediateProof WritingProof Techniques
IntermediateProof WritingDirect Proofs
IntermediateProof WritingIndirect Proofs
IntermediateProof WritingProof by Contradiction
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
AdvancedAnalysisLimits and Continuity
AdvancedAnalysisSequences and Series
AdvancedAlgebraRings 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:

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. April 1 2023.



CC BY-SA 4.0 Jacob Zelko. Last modified: May 24, 2023. Website built with Franklin.jl and the Julia programming language.