Discrete Mathematics with Combinatorics

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.