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This text is a sophomore- or junior-level introduction to the fundamental concepts and techniques used in abstract mathematics. It guides students through the transition from a more computational course to higher level work. It does so by actively engaging students in every step of the development of a central core of ideas common to all branches of mathematics.
FEATURES:
Author Bio
Schumacher, Carol : Kenyon College
Chapter 1: Introduction--an Essay
Mathematical Reasoning
Deciding What to Assume
What Is Needed to Do Mathematics?
Chapter Zero
Chapter 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?
Chapter 3: Sets
Sets and Set Notation
Set Operations
Russell's Paradox
Chapter 4: Relations and Ordering
Relations
Orderings
Equivalence Relations
Chapter 5: Functions
Basic Ideas
Composition and Inverses
Order Isomorphisms
Sequences
Binary Operations
Chapter 6: Induction
Inductive Reasoning and Mathematical Induction
Using Induction
Complete Induction
Chapter 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
Chapter 8: Cardinality
Galileo's Paradox
Infinite Sets
Countable Sets
Beyond Countability
Comparing Cardinalities
The Continuum Hypothesis
Order Isomorphisms (Revisited)
Chapter 9: The Real Numbers
Constructing the Axioms
Arithmetic
Order
The Least Upper Bound Axiom
Sequence Convergence in R
Chapter 10: Axiomatic Set Theory
Elementary Axioms
The Axiom of Infinity
Axioms of Choice and Substitution
Chapter 11: Constructing R
From N to Integers
From Integers to Rationals
From Rationals to R
Index
This text is a sophomore- or junior-level introduction to the fundamental concepts and techniques used in abstract mathematics. It guides students through the transition from a more computational course to higher level work. It does so by actively engaging students in every step of the development of a central core of ideas common to all branches of mathematics.
FEATURES:
Author Bio
Schumacher, Carol : Kenyon College
Table of Contents
Chapter 1: Introduction--an Essay
Mathematical Reasoning
Deciding What to Assume
What Is Needed to Do Mathematics?
Chapter Zero
Chapter 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?
Chapter 3: Sets
Sets and Set Notation
Set Operations
Russell's Paradox
Chapter 4: Relations and Ordering
Relations
Orderings
Equivalence Relations
Chapter 5: Functions
Basic Ideas
Composition and Inverses
Order Isomorphisms
Sequences
Binary Operations
Chapter 6: Induction
Inductive Reasoning and Mathematical Induction
Using Induction
Complete Induction
Chapter 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
Chapter 8: Cardinality
Galileo's Paradox
Infinite Sets
Countable Sets
Beyond Countability
Comparing Cardinalities
The Continuum Hypothesis
Order Isomorphisms (Revisited)
Chapter 9: The Real Numbers
Constructing the Axioms
Arithmetic
Order
The Least Upper Bound Axiom
Sequence Convergence in R
Chapter 10: Axiomatic Set Theory
Elementary Axioms
The Axiom of Infinity
Axioms of Choice and Substitution
Chapter 11: Constructing R
From N to Integers
From Integers to Rationals
From Rationals to R
Index