• An Introduction to Mathematical Proof
  • 1 Introduction
    • 1.1 About this course
    • 1.2 About this book
    • 1.3 How to use this book
    • 1.4 Contents
  • 2 Statements
    • 2.1 Non-statements
    • 2.2 What makes a good statement?
  • 3 Truth values
    • 3.1 What is the truth value of a statement?
    • 3.2 Statements with clearly defined truth values
    • 3.3 Statements without clearly defined truth values
    • 3.4 Unknown truth values
  • 4 Mathematical statements
    • 4.1 Definition
    • 4.2 Mathematical statements don’t need to be about mathematics!
    • 4.3 Statements containing mathematics aren’t necessarily mathematical!
  • 5 Axioms and definitions
    • 5.1 Axioms are a special kind of mathematical statement
    • 5.2 Definitions are a special kind of axiom
    • 5.3 Types of real numbers
    • 5.4 Inequalities
  • 6 Mathematical proof
    • 6.1 Mathematics versus other fields
    • 6.2 So what does this all mean?
  • 7 Disproving a conjecture
    • 7.1 Finding a counterexample
    • 7.2 Finding a disproof
  • 8 Getting started with a new conjecture
    • 8.1 Step 1: Start by looking for a counterexample
    • 8.2 Step 2: Try to build a proof
  • 9 Before we start…
    • 9.1 How do I write a good proof?
    • 9.2 How do I check my proof is correct?
    • 9.3 Key things to bear in mind when proving
    • 9.4 Some useful arrows
    • 9.5 Example
  • 10 The flowchart of proof
  • 11 ▶ Proof by exhaustion
    • 11.1 Steps
    • 11.2 Formal definition
    • 11.3 Conjectures
  • 12 ▶ Direct proof
    • 12.1 Steps
    • 12.2 Formal definition
    • 12.3 Conjectures
  • 13 ▶ Proof by cases
    • 13.1 Steps
    • 13.2 Formal definition
    • 13.3 Exhaustion versus cases
    • 13.4 Conjectures
  • 14 Without loss of generality
    • 14.1 When w.l.o.g. does work
    • 14.2 When w.l.o.g. doesn’t work
  • 15 Maps revisited
    • 15.1 \(\Rightarrow\) revisited
    • 15.2 More relations within a map
    • 15.3 A conjecture and its negation, converse, inverse, and contrapositive
  • 16 ▶ Proof by contrapositive
    • 16.1 Steps
    • 16.2 Formal definition
    • 16.3 Conjectures
  • 17 ▶ Proof by contradiction
    • 17.1 Steps
    • 17.2 Formal definition
    • 17.3 Conjectures
    • 17.4 Methods of contradiction and contrapositive
  • 18 ▶ Proof by induction
    • 18.1 The steps
    • 18.2 Formal definition
    • 18.3 Conjectures
    • 18.4 Going the “other” way
  • 19 ▶ Proof by smallest counterexample
  • 20 ▶ Proof without words
  • 21 ▶ Geometric proof
  • 22 Writing your own conjectures task
  • 23 More conjectures I: Assorted
  • 24 More conjectures II: Prime numbers
    • 24.1 Work through in order, building on your earlier work
    • 24.2 Conjectures
  • 25 Mathematical incompleteness
  • 26 Glossary
  • 27 Representations of numbers
  • 28 List of sources
  • Published with bookdown

An Introduction to Mathematical Proof

Chapter 28 List of sources

Aside from those created by the author, the following were extremely useful in the compiling of conjectures for this book:

https://www.cse.wustl.edu/~cytron/547Pages/f14/IntroToProofs_Final.pdf

http://sites.millersville.edu/bikenaga/math-proof/proof-by-cases/proof-by-cases.pdf

https://www.birmingham.ac.uk/Documents/college-eps/college/stem/Student-Summer-Education-Internships/Proof-and-Reasoning.pdf#page17

http://www.columbia.edu/~md3405/Proofs\%20Solutions.pdf#page11

https://www.cs.colostate.edu/~cs160/.Fall13/assignments/Homework2Solution.pdf

http://mrpaynemath.weebly.com/uploads/3/8/9/9/38994693/math2201ch1.3notes-workings.pdf

http://www.cs.nthu.edu.tw/~wkhon/math/lecture/lecture04.pdf

https://storm.cis.fordham.edu/zhang/cs2100/slides/IntroToProof.pdf

http://people.cs.pitt.edu/~litman/courses/cs441/lecture7.pdf

https://web.maths.unsw.edu.au/~jim/proofsch3.pdf

https://www.people.vcu.edu/~rhammack/Discrete/ChapterDirect.pdf

http://mrpaynemath.weebly.com/uploads/3/8/9/9/38994693/math2201ch1.3notes-workings.pdf

http://alevelmathsrevision.com/downloads/A2/Pure/Proof

https://madasmaths.com/archive/maths_booklets/further_topics/various/proof_general.pdf

https://www.cs.sfu.ca/~ggbaker/zju/math/proof.html

https://www.khanacademy.org/math/geometry-home/geometry-miscellaneous/geometry-worked-examples/v/ca-geometry-proof-by-contradiction