Introductory Concepts for Abstract Mathematics

Introductory Concepts for Abstract Mathematics

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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.show more

Product details

  • Hardback | 344 pages
  • 156 x 232 x 24mm | 619.99g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 2003.
  • 15 black & white tables
  • 1584881348
  • 9781584881346

Review quote

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

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 PROBLEMSshow more