ABSTRACT

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.

part |2 pages

SECTION I LOGIC AND PROOF

chapter 1|12 pages

Logic and Propositional Calculus

chapter 2|14 pages

Tautologies and Validity Chapter

chapter 3|12 pages

Quantifiers and Predicates Chapter

chapter 6|10 pages

On Theorem Proving and Writing Proofs

chapter 7|12 pages

Mathematical Induction

part |2 pages

SECTION II: SETS

chapter 8|14 pages

Sets and Set Operations

chapter 9|16 pages

Set Union, Intersection, and Complement

chapter 10|10 pages

Generalized Union and Intersection

part |2 pages

SECTION III: FUNCTIONS AND RELATIONS

chapter 11|6 pages

Cartesian Products

chapter 12|10 pages

Relations

chapter 13|10 pages

Partitions

chapter 14|12 pages

Functions

chapter 15|6 pages

Composition of Functions

chapter 16|8 pages

Image and Preimage Functions

part |2 pages

SECTION IV: ALGEBRAIC AND ORDER PROPERTIES OF NUMBER SYSTEMS

chapter 17|6 pages

Binary Operations

chapter 18|6 pages

The Systems of Whole and Natural Numbers

chapter 19|14 pages

The System Z of Integers

chapter 20|8 pages

The System Q of Rational Numbers

chapter 21|4 pages

Other Aspects of Order

chapter 22|8 pages

The Real Number System

part |2 pages

SECTION V: TRANSFINITE CARDINAL NUMBERS

chapter 23|6 pages

Finite and Infinite Sets

chapter 24|10 pages

Denumerable and Countable Sets

chapter 25|6 pages

Uncountable Sets

chapter 26|14 pages

Transfinite Cardinal Numbers

part |2 pages

SECTION VI: AXIOM OF CHOICE AND

chapter 27|10 pages

Partially Ordered Sets

chapter 28|4 pages

Least Upper Bound and Greatest Lower Bound

chapter 29|10 pages

Axiom of Choice

chapter 30|6 pages

Well Ordered Sets