1st Edition

Introductory Concepts for Abstract Mathematics

By Kenneth E. Hummel Copyright 2000
    344 Pages
    by Chapman & Hall

    344 Pages
    by Chapman & Hall

    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.

    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

    Biography

    Hummel, Kenneth E.

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