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Introductory Concepts for Abstract Mathematics





ISBN 9780367398835
Published September 5, 2019 by Chapman and Hall/CRC
344 Pages

 
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Book 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.

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

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Author(s)

Biography

Hummel, Kenneth E.

Reviews

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