Synopses & Reviews
This book is designed for the sophomore/junior level Introduction to Advanced Mathematics course. Written in a modified R.L. Moore fashion, it offers a unique approach in which readers construct their own understanding. However, while readers are called upon to write their own proofs, they are also encouraged to work in groups. There are few finished proofs contained in the text, but the author offers “proof sketches” and helpful technique tips to help readers as they develop their proof writing skills. This book is most successful in a small, seminar style class.
Logic, Sets, Induction, Relations, Functions, Elementary Number Theory, Cardinality, The Real Numbers
For all readers interested in abstract mathematics.
Table of Contents
0. Introduction-an Essay
Mathematical Reasoning.
Deciding What to Assume.
What Is Needed to Do Mathematics?
Chapter Zero
1. Logic.
True or False.
Thought Experiment: True or False.
Statements and Predicates.
Quantification.
Mathematical Statements.
Mathematical Implication.
Direct Proofs.
Compound Statements and Truth Tables.
Learning from Truth Tables.
Tautologies.
What About the Converse?
Equivalence and Rephrasing.
Negating Statements.
Existence Theorems.
Uniqueness Theorems.
Examples and Counter Examples.
Direct Proof.
Proof by Contrapositive.
Proof by Contradiction.
Proving Theorems: What Now?
Problems.
Questions to Ponder
2. Sets.
Sets and Set Notation.
Subsets.
Set Operations.
The Algebra of Sets.
The Power Set.
Russell's Paradox.
Problems.
Questions to Ponder.
3. Induction.
Mathematical Induction.
Using Induction.
Complete Induction.
Questions to Ponder.
4. Relations.
Relations.
Orderings.
Equivalence Relations.
Graphs.
Coloring Maps.
Problems.
Questions to Ponder.
5. Functions.
Basic Ideas.
Composition and Inverses.
Images and Inverse Images.
Order Isomorphisms.
Sequences.
Sequences with Special Properties.
Subsequences.
Constructing Subsequences Recursively.
Binary Operations.
Problems.
Questions to Ponder
6. Elementary Number Theory.
Natural Numbers and Integers.
Divisibility in the Integers.
The Euclidean Algorithm.
Relatively Prime Integers.
Prime Factorization.
Congruence Modulo n.
Divisibility Modulo n.
Problems.
Questions to Ponder.
7. Cardinality.
Galileo's Paradox.
Infinite Sets.
Countable Sets.
Beyond Countability.
Comparing Cardinalities.
The Continuum Hypothesis.
Problems.
Questions to Ponder.
8. The Real Numbers.
Constructing the Axioms.
Arithmetic.
Order.
The Least Upper Bound Axiom.
Sequence Convergence in R.
Problems.
Questions to Ponder.
A. Axiomatic Set Theory.
Elementary Axioms.
The Axiom of Infinity.
Axioms of Choice and Substitution.
B. Constructing R.
From N to Z.
From Z to Q.
From Q to R.
Index.