Basic Concepts of Mathematics
Elias Zakon
Basic Concepts of Mathematics
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Description
Contents
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This book helps the student complete the transition from purely manipulative to rigorous mathematics. The clear exposition covers many topics that are assumed by later courses but are often not covered with any depth or organization: basic set theory, induction, quantifiers, functions and relations, equivalence relations, properties of the real numbers (including consequences of the completeness axiom), fields, and basic properties of n-dimensional Euclidean spaces.

The many exercises and optional topics (isomorphism of complete ordered fields, construction of the real numbers through Dedekind cuts, introduction to normed linear spaces, etc.) allow the instructor to adapt this book to many environments and levels of students. Extensive hypertextual cross-references and hyperlinked indexes of terms and notation add truly interactive elements to the text.

There is a list of errata that records changes made to the text since the version of October 29, 2001. Other formats available at trillia.com

Language
English
ISBN
Unknown
Basic Concepts of Mathematics
Copyright Notice
Contents
Preface
About the Author
1 Some Set Theoretical Notions
1 Introduction. Sets and their Elements
2 Operations on Sets
Problems in Set Theory
3 Logical Quantifiers
4 Relations (Correspondences)
Problems in the Theory of Relations
5 Mappings
Problems on Mappings
6 Composition of Relations and Mappings
Problems on the Composition of Relations
7 Equivalence Relations
Problems on Equivalence Relations
8 Sequences
Problems on Sequences
9 Some Theorems on Countable Sets
Problems on Countable and Uncountable Sets
2 The Real Number System
1 Introduction
2 Axioms of an Ordered Field
3 Arithmetic Operations in a Field
4 Inequalities in an Ordered Field. Absolute Values
Problems on Arithmetic Operations and Inequalities in a Field
5 Natural Numbers. Induction
6 Induction (continued)
Problems on Natural Numbers and Induction
7 Integers and Rationals
Problems on Integers and Rationals
8 Bounded Sets in an Ordered Field
9 The Completeness Axiom. Suprema and Infima
Problems on Bounded Sets, Infima, and Suprema
10 Some Applications of the Completeness Axiom
Problems on Complete and Archimedean Fields
11 Roots. Irrational Numbers
Problems on Roots and Irrationals
12 Powers with Arbitrary Real Exponents
Problems on Powers
13 Decimal and other Approximations
Problems on Decimal and q-ary Approximations
14 Isomorphism of Complete Ordered Fields
Problems on Isomorphisms
15 Dedekind Cuts. Construction of E1
Problems on Dedekind Cuts
16 The Infinities. The lim inf and lim sup of a Sequence
Problems on Upper and Lower Limits of Sequences in E*
3 The Geometry of n Dimensions. Vector Spaces
1 Euclidean n-space
Problems on Vectors
2 Inner Products. Absolute Values. Distances
Problems on Vectors (continued)
3 Angles and Directions
4 Lines and Line Segments
Problems on Lines, Angles, and Directions
5 Hyperplanes. Linear Functionals
Problems on Hyperplanes
6 Review Problems on Planes and Lines
7 Intervals
Problems on Intervals
8 Complex Numbers
Problems on Complex Numbers
9 Vector Spaces. The Space Cn. Euclidean Spaces
Problems on Linear Spaces
10 Normed Linear Spaces
Problems on Normed Linear Spaces
Notation
Index
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