Introductory Concepts for Abstract Mathematics

by Hummel; Kenneth E.

• ISBN13: 9781584881346
• ISBN10: 1584881348
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-03-23
• Publisher: Chapman & Hall/

Summary

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 mathematicsThe 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 numbersIntroductory 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

Preface	vii
SECTION I LOGIC AND PROOF	1	(90)
Logic and Propositional Calculus	3	(12)
Tautologies and Validity	15	(14)
Quantifiers and Predicates	29	(12)
Techniques of Derivation and Rules of Inference	41	(12)
Informal Proof and Theorem-Proving Techniques	53	(16)
On Theorem Proving and Writing Proofs	69	(10)
Mathematical Induction	79	(12)
SECTION II SETS	91	(42)
Sets and Set Operations	93	(14)
Set Union, Intersection, and Complement	107	(16)
Generalized Union and Intersection	123	(10)
SECTION III FUNCTIONS AND RELATIONS	133	(54)
Cartesian Products	135	(6)
Relations	141	(10)
Partitions	151	(10)
Functions	161	(12)
Composition of Functions	173	(6)
Image and Preimage Functions	179	(8)
SECTION IV ALGEBRAIC AND ORDER PROPERTIES OF NUMBER SYSTEMS	187	(48)
Binary Operations	189	(6)
The Systems of Whole and Natural Numbers	195	(6)
The Systems Z of Integers	201	(14)
The System Q of Rational Numbers	215	(8)
Other Aspects of Order	223	(4)
The Real Number System	227	(8)
SECTION V TRANSFINITE CARDINAL NUMBERS	235	(38)
Finite and Infinite Sets	237	(6)
Denumerable and Countable Sets	243	(10)
Uncountable Sets	253	(6)
Transfinite Cardinal Numbers	259	(14)
SECTION VI AXIOM OF CHOICE AND ORDINAL NUMBERS	273	(32)
Partially Ordered Sets	275	(10)
Least Upper Bound and Greatest Lower Bound	285	(4)
Axiom of Choice	289	(10)
Well Ordered Sets	299	(6)
Reading List	305	(2)
Hints and Solutions to Selected Exercises	307	(20)
Index	327