[Book Cover]

Discrete Mathematics with Combinatorics, 1/e

James Anderson, University of South Carolina, Spartanburg

Coming March, 2000 by Prentice Hall Engineering/Science/Mathematics

Copyright 2000, 864 pp.
Cloth
ISBN 0-13-086998-8


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Summary

For freshman-level, one- or two-semester courses in Discrete Mathematics. This carefully organized, very readable text covers every essential topic in discrete mathematics in a logical fashion. Placing each topic in context, it covers concepts associated with discrete mathematical systems that have applications in computer science, engineering, and mathematics. The author introduces more basic concepts at the freshman level than are found in other texts, in a simple, accessible form. Introductory material is balanced with extensive coverage of graphs, trees, recursion, algebra, theory of computing, and combinatorics. Extensive examples throughout the text reinforce concepts.

Features


More combinatorics/algebraic structures than in most texts.

  • Removes the necessity of having to go on to take a separate combinatorics course.
Detailed discussion of and strong emphasis on proofs.
  • Helps students understand this essential topic, encouraging them to develop mathematical maturity.
Extensive, in-depth presentation of topics.
  • Enables instructors to select the topics that best meet their goals for the course with plenty of material available.
All key ideas grouped together in the first six chapters (unique to this market).
  • Introduces students to important concepts early; also facilitates a one semester course.
Large selection of applied and computational problems—Ranging from the elementary to the more advanced. More topics in probability and more statistical interpretations than other texts.
  • Helps students with other topics they will encounter in computer science.
Comprehensive discussion of topics such as finite state machines, automata, and languages—Also addresses groups, monoids, lattices, Polish notation, and Karnaugh maps.
Earlier introduction of matrices and relations, Boolean algebras and circuits than most texts.
Includes algorithms for many constructive tasks that occur in discrete systems.
  • Helps facilitate the use of the computer if so desired by the professor.


Table of Contents
    1. Truth Tables, Propositional Logic, and Circuit Diagrams.

      Statements and Connectives. Conditional Statements. Equivalent Statements. Introduction to Axiomatic Systems; Arguments. Completeness in Propositional Logic. Karnaugh Maps. Circuit Diagrams.

    2. Set Theory.

      Introduction to Sets. Set Operations. Venn Diagrams. Boolean Algebras. Relations. Graphs. Directed Graphs. Trees. Partially Ordered Sets. Equivalence Relations. Congruence. Functions. Special Functions. Matrices. Cardinality.

    3. Logic, Number Theory, and Proofs.

      Predicate Calculus. Basic Concepts of Number Theory and Proofs. Mathematical Induction. Divisibility. Prime Integers. Cardinals Revisited.

    4. Algorithms and Recursion.

      The “for” Procedure and Algorithms for Matrix Operations. Prefix and Postfix Notation. Recursive Functions and Algorithms. Complexity of Algorithms. Binary and Hexadecimal Numbers. Signed Numbers.

    5. Algorithms in Number Theory.

      Sieve of Eratosthemes. Fermat's Factorization Method. The Division and Euclidean Algorithms. Integral Solutions of Linear Wquations. Solutions of Congruence Equations. Chinese Remainder Theorem.

    6. Counting and Probability.

      Basic Counting Principles. Permutations, Words, and Arrangements. Combinations and Partitions. Probability. Conditional Probability. Discrete Probability.

    7. Graphs Revisited.

      Special Problems and Graphs. Euler paths and Cycles. Incidence and Adjacency Matrices. Algebraic Properties of Graphs. Planar Graphs. Coloring Graphs. Hamiltonian Graphs. Weighted Graphs and Shortest Path Algorithms.

    8. Trees Revisited.

      Properties of Trees. Binary Search Trees. Weighted Trees. Traversing Binary Trees. Spanning Trees. Minimal Spanning Trees.

    9. Algebraic Structures.

      Partially Ordered Sets Revisited. Semigroups and Semilattices. Lattices. Boolean Algebras. Groups. Rings. Integral Domains. Fields.

    10. Theory of Computations.

      Regular Languages. Automata. Minimal Deterministic Automata and Syntactic Monoids. Kleene's Theorem. Grammars. Pushdown Automata and Context-free Languages. Turing machines. The Halting Problem for Turing Machines.

    11. Recursion Revisited.

      Difference Equations. Homogeneous Linear Recurrence Relations. Nonhomogeneous Linear Recurrence Relations. Generating Functions and Recurrence Relations.


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