Introductory Concepts for Abstract Mathematics: 1st Edition (Hardback) book cover

Introductory Concepts for Abstract Mathematics

1st Edition

By Kenneth E. Hummel

Chapman and Hall/CRC

344 pages

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Hardback: 9781584881346
pub: 2000-03-23
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Description

Beyond calculus, the world of mathematics grows increasingly abstract and places new and challenging demands on those venturing into that realm. As the focus of calculus instruction has become increasingly computational, it leaves many students ill prepared for more advanced work that requires the ability to understand and construct proofs.

Introductory Concepts for Abstract Mathematics helps readers bridge that gap. It teaches them to work with abstract ideas and develop a facility with definitions, theorems, and proofs. They learn logical principles, and to justify arguments not by what seems right, but by strict adherence to principles of logic and proven mathematical assertions - and they learn to write clearly in the language of mathematics

The author achieves these goals through a methodical treatment of set theory, relations and functions, and number systems, from the natural to the real. He introduces topics not usually addressed at this level, including the remarkable concepts of infinite sets and transfinite cardinal numbers

Introductory Concepts for Abstract Mathematics takes readers into the world beyond calculus and ensures their voyage to that world is successful. It imparts a feeling for the beauty of mathematics and its internal harmony, and inspires an eagerness and increased enthusiasm for moving forward in the study of mathematics.

Reviews

"… very clearly written. Sophomore-level undergraduates should have no difficulty with the book."

-Zentralblatt fur Mathematik

Table of Contents

LOGICAL AND PROOF

Logical and Propositional Calculus

Tautologies and Validity

Quantifiers and Predicates

Techniques of Derivation and Rules of Inference

Informal Proof and Theorem-Proving Techniques

On Theorem proving and Writing Proofs

Mathematical Induction

SETS

Sets and Set Operations

Union, Intersection, and Complement

Generalized Union and Intersection

FUNCTIONS AND RELATIONS

Cartesian Products

Relations

Partitions

Functions

Composition of Functions

Image and Preimage Functions

ALGEBRAIC AND ORDER PROPERTIES OF NUMBER SYSTEMS

Binary Operations

The Systems of Whole and Natural Numbers

The System Z of Integers

The System Q of Rational Numbers

Other Aspects of Order

The Real Number System

TRANSFINITE CARDINAL NUMBERS

Finite and Infinite Sets

Denumerable and Countable Sets

Uncountable Sets

Transfinite Cardinal Numbers

AXIOM OF CHOICE AND ORDINAL NUMBERS

Partially Ordered Sets

Least Upper Bound and Greatest Lower Bound

Axiom of Choice

Well Ordered Sets

READING LIST

HINTS AND SOLUTIONS TO SELECTED PROBLEMS

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT028000
MATHEMATICS / Set Theory