Introductory Concepts for Abstract Mathematics

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CRC Press, Mar 23, 2000 - Mathematics - 344 pages
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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.
 

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Contents

Logic and Propositional Calculus
3
Tautologies and Validity
15
Quantifiers and Predicates
29
Techniques of Derivation and Rules
41
Techniques
53
Mathematical Induction
79
Sets and Set Operations
93
Set Union Intersection and Complement
107
Composition of Functions
173
Image and Preimage Functions
179
ALGEBRAIC AND ORDER PROPERTIES
187
Finite and Infinite Sets
237
Denumerable and Countable Sets
243
Uncountable Sets
253
Transfinite Cardinal Numbers
259
Partially Ordered Sets
275

Generalized Union and Intersection
123
Cartesian Products
135
Relations
141
Partitions
151
Functions
161
Least Upper Bound and Greatest Lower
285
Well Ordered Sets
299
READING LIST
305
INDEX
327
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