Methods of Proof
Instructor: Dr. Dylan Retsek
Oce: 314 Faculty Oces East
email: [email protected]
Web: www.calpoly.edu/ dretsek
Phone: (805) 756-2072
Our objective this quarter is to fundamentally recast how we thin
The Promised Land
I am the master of my fate: I am the captain of my soul.
From Invictus, by William Ernest Henley
This chapter is devoted to the concept of function, king of all relations. The
fact that you have experience with functions can be
Use punctuation, its classy.
What Constitutes Proof ?
In Section 1.1 we spent a fair bit of time convincing one another of the truth or
falsity of eight propositions. What followed in chapter one was all the formalism
CHAPTER 1. THE WORDS MATTER
These truth tables are designed to succinctly summarize denitions 1.18-1.20.
To wit, the rst truth table above merely indicates that the conjunction P Q is
true only when both constituent propositions P and Q are true.
CHAPTER 2. MATHEMATICAL PROOF
Proofs of Quantied Propositions
Many of the most fundamental denitions and theorems in advanced mathematics
involve the quantiers and . Facility in proving quantied propositions is critical
to future success.
The Theory of Sets
Your proof is not merely a demonstration, its also a narrative.
The Stage is Sets
Denition 3.1. A set is a collection of objects called its elements. If A is a set and
x is an element of A, we write x A. Otherwise, we writ
The Stu of Legends
The denitions shall set you free.
This chapter is the primordial soup from which much of modern mathematics
springs. Keep a lookout for the vestiges of some of your most familiar and cherished
mathematical objects and savor th
Indexed Families of Sets
After our lengthy discussion on indexed families, it seems some follow up is
in order. The following exercises are not ever due and need not be included in
your portfolios. They are just for you to work on to
The Words Matter
If you dont say it well, people wont know what youre talking about.
We begin with a denition.
Denition 1.1. The length of a word is however many letters long it is. Given a
word W , denote its length by l(W ).