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

ISBN-10: 0130869988
ISBN-13: 9780130869982

Publisher: Pearson
Copyright: 2001
Format: Cloth; 799 pp
Published: 07/20/2000

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.

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. Ex.___

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. Ex.___

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. Ex.___

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. Ex.___

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.

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 Continued.

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.

10. Number Theory Revisited.

Integral Solutions of Linear Equations. Solutions of Congruence Equations. Chinese Remainder Theorem. Properties of the Function. 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. Grammars.

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.

Discrete Math (Mathematics)

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