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Chapter Zero : Fundamentals Notions of Abstract Mathematics

Chapter Zero : Fundamentals Notions of Abstract Mathematics - 96 edition

ISBN13: 978-0201826531

Cover of Chapter Zero : Fundamentals Notions of Abstract Mathematics 96 (ISBN 978-0201826531)
ISBN13: 978-0201826531
ISBN10: 0201826534
Cover type:
Edition/Copyright: 96
Publisher: Addison-Wesley Longman, Inc.
Published: 1996
International: No

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Chapter Zero : Fundamentals Notions of Abstract Mathematics - 96 edition

ISBN13: 978-0201826531

Carol Schumacher

ISBN13: 978-0201826531
ISBN10: 0201826534
Cover type:
Edition/Copyright: 96
Publisher: Addison-Wesley Longman, Inc.

Published: 1996
International: No
Summary

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:

  • Following the foundation work of Chapters 1­6, the remaining, self-contained, chapters offer three alternative avenues for continued study. This text uses a unique approach in which students construct their own understandings.
  • Logic is used as a tool for analyzing the content of mathematical assertions and for constructing valid mathematical proofs.
  • Order structure is used primarily in Chapter 4 and extensively throughout the rest of the text.
  • Rigorous axiomatic treatment of set theory is introduced in Appendices A and B (which are written in the same style as the text's chapters).

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

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