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Discrete Mathematics with Combinatorics, 2/E
James A. Anderson, University of South Carolina-Spartanburg

ISBN-10: 0130457914
ISBN-13:  9780130457912

Publisher:  Pearson
Copyright:  2004
Format:  Paper; 928 pp
Published:  08/19/2003
Status: Out of Print


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Description

For one-/two- semester, freshman 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

  • NEW - Expanded coverage throughout on writing mathematical proof.

    • Supplies students with a more comprehensive and in-depth presentation of discrete mathematics topics.

  • NEW - Extensive increase in problems in Chapters 1-8—Includes an increase in the number that involve proofs.

    • Helps students to think logically and understand the basic concepts before moving on to more advanced material.

  • NEW - Increased coverage of theory of computation.

    • Provides students with material on finite states with output, grammars, and Turing machines.

  • NEW - End of chapter summaries and quizzes.

    • Offers useful study aids to students.

  • NEW - Boxed off computer applications for computer science applications.

    • Makes the text more appealing to CS students.

  • Combinatorics/algebraic structures—Covered more extensively 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.

  • Unique format—Groups all key ideas together in the first eight chapters (unique to this market).

    • Introduces students to important concepts early; also facilitates a one-semester course.

  • Large selection of applied and computational problems—Ranges from the elementary to the more advanced, with more topics in probability and more statistical interpretations than found in other texts.

    • Helps students grasp other topics they will encounter in computer science.

  • Comprehensive discussion of CS 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 in most texts.

    • Presents students, in later chapters, with expanded discussion of previously covered topics and more advanced ones.

  • Algorithms for many constructive tasks that occur in discrete systems.

    • Encourages the use of the computer if so desired by the professor.


New To This Edition

  • Expanded coverage throughout on writing mathematical proof.

    • Supplies students with a more comprehensive and in-depth presentation of discrete mathematics topics.

  • Extensive increase in problems in Chapters 1-8—Includes an increase in the number that involve proofs.

    • Helps students to think logically and understand the basic concepts before moving on to more advanced material.

  • Increased coverage of theory of computation.

    • Provides students with material on finite states with output, grammars, and Turing machines.

  • End of chapter summaries and quizzes.

    • Offers useful study aids to students.

  • Boxed off computer applications for computer science applications.

    • Makes the text more appealing to CS students.


Table of Contents



1. Truth Tables, Logic, and Proofs.

Statements and Connectives. Conditional Statements. Equivalent Statements. Axiomatic Systems: Arguments and Proofs. Completeness in Propositional Logic. Karnaugh Maps. Circuit Diagrams.



2. Set Theory.

Introduction to Sets. Set Operations. Venn Diagrams. Boolean Algebras. Relations. Partially Ordered Sets. Equivalence Relations. Functions.



3. Logic, Integers, and Proofs.

Predicate Calculus. Basic Concepts of Proofs and the Structure of Integers. Mathematical Induction. Divisibility. Prime Integers. Congruence Relations.



4. Functions and Matrices.

Functions. Special Functions. Matrices. Cardinality. Cardinals Revisited.



5. Algorithms and Recursion.

The “for” Procedure and Algorithms for Matrices. Recursive Functions and Algorithms. Complexity of Algorithms. Sorting Algorithms. Prefix and Suffix Notation. Binary and Hexadecimal Numbers. Signed Numbers. Matrices Continued.



6. Graphs, Directed Graphs and Trees.

Graphs. Directed Graphs. Trees. Instant Insanity. Euler Paths and Cycles. Incidence and Adjacency Matrices. Hypercubes and Gray Code.



7. Number Theory.

Sieve of Eratosthenes. Fermat's Factorization Method. The Division and Euclidean Algorithms. Continued Fractions. Convergents.



8. Counting and Probability.

Basic Counting Principles. Inclusion-Exclusion Introduced. Permutations and Combinations. Generating Permutations and Combinations. Probability Introduced. Generalized Permutations and Combinations. Permutations and Combinations with Repetition. Pigeonhole Principle. Probability Revisited. Bayes' Theorem. Markov Chains.



9. Algebraic Structures.

Partially Ordered Sets Revisited. Semigroups and Semilattices. Lattices. Groups. Groups and Homomorphisms. Linear Algebra.



10. Number Theory Revisited.

Integral Solutions of Linear Equations. Solutions of Congruence Equations. Chinese Remainder Theorem. Order of an Integer.



11. Recursion Revisited.

Homogeneous Linear Recurrence Relations. Nonhomogeneous Linear Recurrence Relations. Finite Differences. Factorial Polynomials. Sums of Differences.



12. Counting Continued.

Occupancy Problems. Catalan Numbers. General Inclusion-Exclusion and Derangements. Rook Polynomials and Forbidden Positions.



13. Generating Functions.

Defining the Generating Function (optional). Generating Functions and Recurrence Relations. Generating Functions and Counting. Partitions. Exponential Generating Functions.



14. Graphs Revisited.

Algebraic Properties of Graphs. Planar Graphs. Coloring Graphs. Hamiltonian Paths and Cycles. Weighted Graphs and Shortest Path Algorithms.



15. Trees.

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



16. Networks.

Networks and Flows. Matching. Petri Nets.



17. Theory of Computation.

Regular Languages. Automata. Finite State Machines with Output. Grammars. Turing Machines.



18. Theory of Codes.

Introduction. Generator Matrices. Hamming Codes.



19. Enumeration of Colors.

Burnside's Theorem. Polya's Theorem.



20. Rings, Integral Domains, and Fields.

Rings and Integral Domains. Integral Domains. Polynomials. Algebra and Polynomials.



21. Group and Semigroup Characters.

Complex Numbers. Group Characters. Semigroup Characters.



22. Applications of Number Theory.

Application: Pattern Matching. Application: Hashing Functions. Application: Cryptography.



Bibliography.


Hints and Solutions to Selected Exercises.


Index.



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