Discrete mathematics

Description

(Each Chapter, excluding Chapter 7, concludes with "Historical Notes," "Supplement Exercises," "Computer Projects," and "Suggested Readings."). 1. An Introduction to Combinatorial Problems and Techniques. The Time to Complete a Project. A Matching Problem. A Knapsack Problem. Algorithms and Their Efficiency. 2. Sets, Relations, and Functions. Set Operations. Equivalence Relations. Congruence. Partial Ordering Relations. Functions. Mathematical Induction. Applications. 3. Graphs. Graphs and Their Representations. Paths and Circuits. Shortest Paths and Distance. Coloring a Graph. Directed Graphs and Multi Graphs. 4. Trees. Properties of Trees. Spanning Trees. Depth-First Search. Rooted Trees. Binary Trees and Traversals. Optimal Binary Trees and Binary Search Trees. 5. Matching. Systems of Distinct Representatives. Matching in Graphs. A Matching Algorithm. Applications of the Algorithm. The Hungarian Method. 6. Network Flows. Flows and Cuts. A Flow Augmentation Algorithm. The Max-Flow Min-Cut Theorem. Flows and Matchings. 7. Counting Techniques. Pascal's Triangle and the Binomial Theorem. Three Fundamental Principles. Permutations and Combinations. Arrangements and Selections with Repetitions. Probability. *The Principle of Inclusion-Exclusion. *Generating Permutations and r-Combinations. 8. Recurrence Relations and Generating Functions. Recurrence Relations. The Method of Iteration. Linear Difference Equations with Constant Coefficients. *Analyzing the Efficiency of Algorithms with Recurrence Relations. Counting with Generating Functions. The Algebra of Generating Functions. 9. Combinatorial Circuits and Finite State Machines. Logical Gates. Creating Combinatorial Circuits. Karnaugh Maps. Finite State Machines. Appendix A: An Introduction to Logic and Proof. Statements and Connectives. Logical Equivalence. Methods of Proof. Appendix B: Matrices. Appendix C: The Algorithms in this Book. Bibliography. Answers to Odd-Numbered Exercises. Index. *Optional Sections