Discrete Mathematics

(NOTE: All Chapters end with Notes, Chapter Review, Chapter Self-Test, and Computer Exercises.)

Preface.


1.  Logic and Proofs.




Propositions. Conditional Propositions and Logical Equivalence. Quantifiers. Proofs. Resolution Proofs. Mathematical Induction. Strong Form of Induction and well ordering Property. 

2. The Language of Mathematics



Sets. Functions. Sequences and Strings. Relations. 



3. Relations.


Relations. Equivalence Relations. Matrices of Relations. Relational Databases.



4. Algorithms.


Introduction. Correctness of Algorithms. Analysis of Algorithms. Recursive Algorithms.



5. Introduction to Number Theory.


Divisors. Representation of Integers and Integer Algorithims. The Euclidean Algorithm. The RSA Public-Key Cryptosystem.



6. Counting Methods and the Pigeonhole Principle.


Basic Principles. Permutations and Combinations. Algorithms for Generating Permutations and Combinations. Introduction to Discrete Probability. Discrete Probability Theory. Generalized Permutations and Combinations. Binomial Coefficients and Combinatorial Identities. The Pigeonhole Principle.  



7. Recurrence Relations.


Introduction. Solving Recurrence Relations. Applications to the Analysis of Algorithms.



8. Graph Theory.


Introduction. Paths and Cycles. Hamiltonian Cycles and the Traveling Salesperson Problem. A Shortest-Path Algorithm. Representations of Graphs. Isomorphisms of Graphs. Planar Graphs. Instant Insanity.



9. Trees.


Introduction. Terminology and Characterizations of Trees. Spanning Trees. Minimal Spanning Trees. Binary Trees. Tree Traversals. Decision Trees and the Minimum Time for Sorting. Isomorphisms of Trees. Game Trees.



10. Network Models.


Introduction. A Maximal Flow Algorithm. The Max Flow, Min Cut Theorem. Matching.



11. Boolean Algebras and Combinatorial Circuits.


Combinatorial Circuits. Properties of Combinatorial Circuits. Boolean Algebras. Boolean Functions and Synthesis of Circuits. Applications



12. Automata, Grammars, and Languages.


Sequential Circuits and Finite-State Machines. Finite-State Automata. Languages and Grammars. Nondeterministic Finite-State Automata. Relationships Between Languages and Automata.



13. Computational Geometry.


The Closest-Pair Problem. An Algorithm to Compute the Convex Hull.



Appendices.


Matrices. Algebra Review.



References.


Hints and Solutions to Selected Exercises.


Index.