• 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 21 ▶ Geometric proof

See https://www.khanacademy.org/math/geometry-home/geometry-miscellaneous/geometry-worked-examples/v/ca-geometry-proof-by-contradiction for more information about geometric proof.