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 20
▶ Proof without words
See
https://artofproblemsolving.com/wiki/index.php/Proofs_without_words
,
http://proofwords.blogspot.com
for more information about the method of
proof without words
.