## A Transition to Advanced Mathematics

### by Professor Steven Roman

These notes (no exercises) accompany my YouTube lecture series with the same title.

I am offering a PDF version of these notes for \$20.00 (plus tax if you live in California). You can pay through PayPal by clicking the link below. AFTER PAYING, PLEASE CLICK 'RETURN TO MERCHANT' ON THE PAYPAL WEB SITE AND YOU WILL BE TAKEN TO A PAGE WHERE YOU CAN DOWNLOAD THE NOTES. • Chapter 1: Introduction
• First Comes Logic
• The Foundations of Modern Mathematics
• The Zermelo-Fraenkel Axioms
• Branches of Mathematics
• Four Fundamental Concepts
• Mathematics Requires Precision
• Unsolved Problems
• Defining the Natural Numbers
• The Axiom of Choice
• Sizes of Infinity
• Some Commonly Used Terms
• Conclusion to the Introduction
• Chapter 2: The Propositional Calculus
• Introduction
• Statements
• Symbolic Form
• Truth Tables
• Constructing Truth Tables
• Logical Equivalence
• Relatives of the Conditional
• Well-Formed Formulas
• Valid Arguments
• Normal Forms
• Logic Circuits
• Chapter 3: The Predicate Calculus
• Quantifiers
• The Language of Predicate Calculus
• The Scope of a Quantifier
• Free and Bound Variables
• Rules for Translating Quantifiers
• Multiple Quantifiers Occur Often
• The Negation of Quantified Statements
• Expressing Uniqueness
• The Axioms of Zermelo-Fraenkel Set Theory
• Chapter 4: Proof Techniques
• Introduction
• Content Versus Style
• What is a Proof?
• Handling Proofs of Conditional Statements
• Proof Templates
• Direct Proof
• Proof By Contraposition
• Proofs Of Biconditional Statements
• Chapter 5: Set Theory
• Set Operations
• Venn Diagrams
• Properties of Set Operations
• Ordered Pairs
• The Cartesian Product
• Multisets
• Families of Sets
• The Union and Intersection of Family of Sets
• Chapter 6: Induction
• Summation Notation
• Mathematical Induction
• The Well-Ordering Principle
• Chapter 7: Combinatorics: The Art of Counting
• The Sum Rule
• The Principle of Inclusion-Exclusion
• The Fundamental Counting Principle
• Permutations
• Combinations
• Poker
• Sperner Families of Sets
• Chapter 8: Binary Relations
• The Definition
• The Graph of a Binary Relation
• The Inverse of a Binary Relation
• Other Types of Relations
• Properties of Binary Relations
• Chapter 9: Partitions and Equivalence Relations
• Partitions
• Equivalence Relations
• Chapter 10: Partially Ordered Sets
• Partially Ordered Sets
• Zorn's Lemma
• Lattices
• Chapter 11: Functions
• The Definition
• Special Types of Functions
• Sequences
• Composition of Functions
• Restrictions and Extensions of Functions
• Surjective Injective and Bijective Functions
• Induced Direct and Inverse Functions
• Increasing and Decreasing Functions
• Chapter 12: Cardinality
• Preliminaries
• Cantor's Insight
• Equipollence
• A Closer Look at Cardinal Numbers
• Finite and Infinite Sets
• Manipulating Images
• Finite Sets
• Countable Sets
• The Power of the Continuum
• The Ordering of Cardinal Numbers
• Other Cardinalities
• The Continuum Hypothesis
• The Arithmetic of Infinite Cardinal Numbers
• The Formal Definition of a Cardinal Number

Home