Math Doctor Bob

Math Major Basics

BM1. Propositional Logic

BM2. Logical Equivalence

BM3. Formal Proofs

BM4. Methods of Proof

BM4.1. Example of Complete Induction

BM5. Naive Set Theory

BM6. Set Operations

BM6.1. Two Tough Set Equalities

BM7. Binary Relations

BM8.1. Mappings: Examples

BM8.2. Mappings: Basic Properties

BM8.3. Mappings: Composition and Inverse Functions

BM9.1. Cardinality 1: Finite Sets

BM9.2. Cardinality 2: Ininite Sets

BM10: Basic Properties of the Integers

BM10.1. The Euclidean Algorithm for the Integers

Complex Analysis

Multiplying Complex Numbers

Complex Conjugate Roots of a Real Polynomial

Factoring 3x^4-2x^3+3x-2 Over The Complex Numbers

Overview of Complex Numbers (1 of 4)

Overview of Complex Numbers (2 of 4)

Overview of Complex Numbers (3 of 4)

Overview of Complex Numbers (4 of 4)

Example of De Moivre's Theorem

Cube Roots of -8

Solutions of z^4-4z^2+16=0

Locus of |z| = |z+2+2i|

Largest Argument of Circle |z+4+2i|

Locus of arg(z-2) = arg(z+2) + pi/3

Locus of Complex equation Z^2 = 1+z/1-z for |z|=1

Parametrizing Curves in the Complex Plane 1

Parametrizing Curves in the Complex Plane 2

Integral of 1/(1+x^4)

Integral of 1/(1+x^{2N})

Integral of 1/(1+sin^2(t))

Real Analysis

Example of Proof by Induction: Increasing Sequence (Monotone Convergence Theorem)

Example of Countable Partition

Cardinality Example with [0,1]

Example of Cauchy Sequence 1

Example of Cauchy Sequence 2

Measure Theory - Motivation

Point Set Topology

Hausdorff Example 1: Cofinite Topology

Hausdorff Example 2: Quotient Space

Hausdorff Example 3: Function Spaces