Advanced Calculus Single Variable

3.3 Equivalence Relations

There are many ways to compare elements of a set other than to say two elements are equal or the same. For example, in the set of people let two people be equivalent if they have the same weight. This would not be saying they were the same person, just that they weighed the same. Often such relations involve considering one characteristic of the elements of a set and then saying the two elements are equivalent if they are the same as far as the given characteristic is concerned.

Definition 3.3.1 Let S be a set. ∼ is an equivalence relation on S if it satisfies the following axioms.

  1. x ∼ x   for all x ∈ S. (Reflexive)
  2. If x ∼ y then y ∼ x. (Symmetric)
  3. If x ∼ y and y ∼ z, then x ∼ z. (Transitive)

Definition 3.3.2

denotes the set of all elements of S which are equivalent to x and
is called the equivalence class determined by x or just the equivalence class of x.

With the above definition one can prove the following simple theorem.

Theorem 3.3.3 Let ∼ be an equivalence class defined on a set, S and let ℋ denote the set of equivalence classes. Then if

and
are two of these equivalence classes, either x ∼ y and
=
or it is not true that x ∼ y and
∩
= ∅.

Chapter August 25, 2018 1 Introduction

Chapter 2 The Real And Complex Numbers

2.1 Real and Rational Numbers
2.2 Exercises
2.3 Set Notation
2.4 Order
2.5 Exercises
2.6 The Binomial Theorem
2.7 Well Ordering Principle And Archimedean Property
2.8 Arithmetic of Integers
2.9 Exercises
2.10 Completeness of ℝ
2.11 Existence Of Roots
2.12 Exercises
2.13 The Complex Numbers
2.14 The Cauchy Schwarz Inequality
2.15 Integer Multiples of Irrational Numbers
2.16 Exercises

Chapter 3 Set Theory

3.1 Basic Definitions
3.2 The Schroder Bernstein Theorem
3.3 Equivalence Relations
3.4 Exercises

Chapter 4 Functions And Sequences

4.1 General Considerations
4.2 Sequences
4.3 Exercises
4.4 The Limit Of A Sequence
4.5 The Nested Interval Lemma
4.6 Exercises
4.7 Compactness
4.7.1 Sequential Compactness
4.7.2 Closed And Open Sets
4.7.3 Compactness And Open Coverings
4.8 Exercises
4.9 Cauchy Sequences And Completeness
4.9.1 Decimals
4.9.2 lim sup and lim inf
4.9.3 Shrinking Diameters
4.10 Exercises

Chapter 5 Infinite Series Of Numbers

5.1 Basic Considerations
5.1.1 Absolute Convergence
5.2 Exercises
5.3 More Tests For Convergence
5.3.1 Convergence Because Of Cancellation
5.3.2 Ratio And Root Tests
5.4 Double Series
5.5 Exercises

Chapter 6 Continuous Functions

6.1 Equivalent Formulations Of Continuity
6.2 Exercises
6.3 The Extreme Values Theorem
6.4 The Intermediate Value Theorem
6.5 Connected Sets
6.6 Exercises
6.7 Uniform Continuity
6.8 Exercises
6.9 Sequences And Series Of Functions
6.10 Sequences Of Polynomials, Weierstrass Approximation
6.11 Ascoli Arzela Theorem∗
6.12 Exercises

Chapter 7 The Derivative

7.1 Limit Of A Function
7.2 Exercises
7.3 The Definition Of The Derivative
7.4 Continuous And Nowhere Differentiable
7.5 Finding The Derivative
7.6 Local Extreme Points
7.7 Exercises
7.8 Mean Value Theorem
7.9 Exercises
7.10 Derivatives Of Inverse Functions
7.11 Derivatives And Limits Of Sequences
7.12 Exercises

Chapter 8 Power Series

8.1 Functions Defined In Terms Of Series
8.2 Operations On Power Series
8.3 The Special Functions Of Elementary Calculus
8.3.1 The Functions, sin,cos,exp
8.3.2 ln and log b
8.4 The Binomial Theorem
8.5 Exercises
8.6 L’Hôpital’s Rule
8.6.1 Interest Compounded Continuously
8.7 Exercises
8.8 Multiplication Of Power Series
8.9 Exercises
8.10 The Fundamental Theorem Of Algebra
8.11 Some Other Theorems

Chapter 9 Integration

9.1 Fundamental Definitions and Properties
9.2 The Fundamental Theorem of Calculus
9.3 Uniform Convergence And The Integral
9.4 A Simple Procedure For Finding Integrals
9.5 Stirling’s Formula
9.6 The Gamma Function
9.7 Laplace Transforms
9.8 Exercises

Chapter 10 Integration On Rough Paths∗

10.1 Finite p Variation
10.2 Piecewise Linear Approximation
10.3 The Young Integral

Chapter 11 Fourier Series

11.1 The Complex Exponential
11.2 Definition And Basic Properties
11.3 The Riemann Lebesgue Lemma
11.4 Dini’s Criterion For Convergence
11.5 Integrating And Differentiating Fourier Series
11.6 Ways Of Approximating Functions
11.6.1 Uniform Approximation With Trig. Polynomials
11.6.2 Mean Square Approximation
11.7 Exercises

Chapter 12 The Generalized Riemann Integral

12.1 Definitions And Basic Properties
12.2 Integrals Of Derivatives
12.3 Exercises
B.1 Closed and Open Sets
B.2 Compactness
B.2.1 Continuous Functions
B.2.2 Convergent Sequences
B.2.3 Continuity and the Limit of a Sequence
B.2.4 The Extreme Value Theorem and Uniform Continuity
B.2.5 Equivalence Of Norms
B.3 Norms On Linear Maps
B.4 Connected Sets
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