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Chapter Zero 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 students construct their own understandings. However, while students 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 students as they develop their proof writing skills. This book is most successful in a small, seminar style class.
Author Bio
Schumacher, Carol : Kenyon College
1. Introduction--an Essay
Mathematical Reasoning
Deciding What to Assume
What Is Needed to Do Mathematics?
Chapter Zero
2. Logic
Statements and Predicates
Mathematical Implication
Direct Proofs
Compound Statements and Truth Tables
Equivalence
Proof by Contrapositive
Negating Statements
Proof by Contradiction
Existence and Uniqueness
Proving Theorems: What Now?
3. Sets
Sets and Set Notation
Set Operations
Russell's Paradox
4. Relations and Ordering
Relations
Orderings
Equivalence Relations
5. Functions
Basic Ideas
Composition and Inverses
Order Isomorphisms
Sequences
Binary Operations
6. Induction
Inductive Reasoning and Mathematical Induction
Using Induction
Complete Induction
7. 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
8. Cardinality
Galileo's Paradox
Infinite Sets
Countable Sets
Beyond Countability
Comparing Cardinalities
The Continuum Hypothesis
Order Isomorphisms (Revisited)
9. The Real Numbers
Constructing the Axioms
Arithmetic
Order
The Least Upper Bound Axiom
Sequence Convergence in Real Numbers
A. Axiomatic Set Theory
Elementary Axioms
The Axiom of Infinity
Axioms of Choice and Substitution
B. Constructing R
From Natural Numbers to Integers
From Integers to Rational Numbers
From Rational Real Numbers to Real Numbers
Index
Chapter Zero 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 students construct their own understandings. However, while students 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 students as they develop their proof writing skills. This book is most successful in a small, seminar style class.
Author Bio
Schumacher, Carol : Kenyon College
Table of Contents
1. Introduction--an Essay
Mathematical Reasoning
Deciding What to Assume
What Is Needed to Do Mathematics?
Chapter Zero
2. Logic
Statements and Predicates
Mathematical Implication
Direct Proofs
Compound Statements and Truth Tables
Equivalence
Proof by Contrapositive
Negating Statements
Proof by Contradiction
Existence and Uniqueness
Proving Theorems: What Now?
3. Sets
Sets and Set Notation
Set Operations
Russell's Paradox
4. Relations and Ordering
Relations
Orderings
Equivalence Relations
5. Functions
Basic Ideas
Composition and Inverses
Order Isomorphisms
Sequences
Binary Operations
6. Induction
Inductive Reasoning and Mathematical Induction
Using Induction
Complete Induction
7. 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
8. Cardinality
Galileo's Paradox
Infinite Sets
Countable Sets
Beyond Countability
Comparing Cardinalities
The Continuum Hypothesis
Order Isomorphisms (Revisited)
9. The Real Numbers
Constructing the Axioms
Arithmetic
Order
The Least Upper Bound Axiom
Sequence Convergence in Real Numbers
A. Axiomatic Set Theory
Elementary Axioms
The Axiom of Infinity
Axioms of Choice and Substitution
B. Constructing R
From Natural Numbers to Integers
From Integers to Rational Numbers
From Rational Real Numbers to Real Numbers
Index