[Book Cover]

Discrete Mathematical Structures, 4/e

Bernard Kolman
Robert Busby, both of Drexel University
Sharon Ross, Georgia Perimiter College

Coming November, 1999 by Prentice Hall Engineering/Science/Mathematics

Copyright 2000, 550 pp.
Cloth
ISBN 0-13-083143-3


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Summary

For one/two-term, freshman/sophomore-level courses in Discrete Mathematics. More than any other book in the field, this text ties together discrete topics with a theme. Written at an appropriate level of rigor—with a strong pedagogical focus—it limits depth of coverage and areas covered to topics of genuine use in computer science. An emphasis on both basic theory and applications provides students with a firm foundation for more advanced courses.

Features


NEW— More emphasis on proofs—Includes a “Tips on Proofs” unit in each chapter. The new unit discusses the proof techniques associated with the topics of that chapter.

  • Offers students more dialogue about reading, understanding, and constructing proofs.
NEW—Extensively redesigned exercise sets—More routine exercises as well as more challenging problems are given. The dialogue on proof and proof techniques continues here with exercises that require students to complete partial proofs and analyze proof strategies. More exercises require explanations or justifications.
NEW—Chapter Tests—Solutions to all questions supplied.
  • Gives students the opportunity to self test their knowledge of chapter concepts.
NEW—The different ways of representing relations are more thoroughly exploited.
  • Enables students with different learning styles to master the concepts.
NEW—Additional sections in the graph theory chapter covering transport networks and their applications.
Still the briefest text on the market. An accessible, concise, and interesting narrative—Devoid of excessive technical jargon and abstraction.
  • Helps students focus on fundamental concepts without becoming bogged down in special cases, weakly motivated examples, and applications. Enables instructors to sharpen focus on key ideas and concepts.
Experiments—Suitable for individual work or group projects.
  • Gives students the opportunity to work with an open-ended problem, to develop and test conjectures, communicate their work and build confidence in their problem-solving skills. Experiments also foster better conceptual understanding.
Only a limited background in mathematics required—No calculus.
  • Offers students a level of rigor appropriate for beginners.
Algorithms and pseudocode used throughout the text to illustrate techniques.
Systematic approach—Introduces each new concept using previously encountered material.
  • Provides students with material developed in such a way to simplify the more complex ideas that follow and lead naturally to other algebraic structures.
Coding exercises—Provided in each chapter.
  • Relate discrete mathematics topics to students' programming experience. Can also be used in developing programming ideas and skills before a programming course.
Organization of topics around the concept of a relation—Treats relations and digraphs as two aspects of the same basic mathematical idea.
  • This fundamental idea is then used as the basis for all the concepts introduced—including functions, partial orders, graphs, and algebraic structures.
A focus on topics of genuine use in computer science.
  • Limits coverage of abstract algebra and gives applications to the important topics of finite state machines, and error correcting and detecting codes.


Table of Contents
    1. Fundamentals.

      Sets and Subsets. Operations on Sets. Sequences. Division in the Integers. Matrices. Mathematical Structures.

    2. Logic.

      Propositions and Logical Operations. Conditional Statements. Methods of Proof. Mathematical Induction.

    3. Counting.

      Permutations. Combinations. The Pigeonhole Principle. Elements of Probability. Recurrence Relations.

    4. Relations and Digraphs.

      Product Sets and Partitions. Relations and Digraphs. Paths in Relations and Digraphs. Properties of Relations. Equivalence Relations. Computer Representation of Relations and Digraphs. Operations on Relations. Transitive Closure and Warshall's Algorithm.

    5. Functions.

      Functions. Functions for Computer Science. Growth of Functions. Permutation Functions.

    6. Order Relations and Structures.

      Partially Ordered Sets. Extremal Elements of Partially Ordered Sets. Lattices. Finite Boolean Algebras. Functions on Boolean Algebras. Circuit Design.

    7. Trees.

      Trees. Labeled Trees. Tree Searching. Undirected Trees. Minimal Spanning Trees.

    8. Topics in Graph Theory.

      Graphs. Euler Paths and Circuits. Hamiltonian Paths and Circuits. Transport Networks. Matching Problems. Coloring Graphs.

    9. Semigroups and Groups.

      Binary Operations Revisited. Semigroups. Products and Quotients of Semigroups. Groups. Products and Quotients of Groups.

    10. Languages and Finite-State Machines.

      Languages. Representations of Special Languages and Grammars. Finite-State Machines. Semigroups, Machines, and Languages. Machines and Regular Languages. Simplification of Machines.

    11. Groups and Coding.

      Coding of Binary Information and Error Detection. Decoding and Error Correction.

    Appendix A: Algorithms and Pseudocode.
    Appendix B: Experiments in Discrete Mathematics.


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