Discrete and Combinatorial Mathematics

PART 1. FUNDAMENTALS OF DISCRETE MATHEMATICS.

1. Fundamental Principles of Counting.
 

The Rules of Sum and Product.  

 

Permutations.  

 

Combinations: The Binomial Theorem.  

 

Combinations with Repetition.  

 

The Catalan Numbers (Optional).  

 

Summary and Historical Review.  

2. Fundamentals of Logic.
 

Basic Connectives and Truth Tables.  

 

Logical Equivalence: The Laws of Logic.  

 

Logical Implication: Rules of Inference.  

 

The Use of Quantifiers.  

 

Quantifiers, Definitions, and the Proofs of Theorems.  

 

Summary and Historical Review.  

3. Set Theory.
 

Sets and Subsets.  

 

Set Operations and the Laws of Set Theory.  

 

Counting and Venn Diagrams.  

 

A First Word on Probability.  

 

The Axioms of Probability (Optional).  

 

Conditional Probability: Independence (Optional).  

 

Discrete Random Variables (Optional).  

 

Summary and Historical Review.  

4. Properties of the Integers: Mathematical Induction.
 

The Well-Ordering Principle: Mathematical Induction.  

 

Recursive Definitions.  

 

The Division Algorithm: Prime Numbers.  

 

The Greatest Common Divisor: The Euclidean Algorithm.  

 

The Fundamental Theorem of Arithmetic.  

 

Summary and Historical Review.  

5. Relations and Functions.
 

Cartesian Products and Relations.  

 

Functions: Plain and One-to-One.  

 

Onto Functions: Stirling Numbers of the Second Kind.  

 

Special Functions.  

 

The Pigeonhole Principle.  

 

Function Composition and Inverse Functions.  

 

Computational Complexity.  

 

Analysis of Algorithms.  

 

Summary and Historical Review.  

6. Languages: Finite State Machines.
 

Language: The Set Theory of Strings.  

 

Finite State Machines: A First Encounter.  

 

Finite State Machines: A Second Encounter.  

 

Summary and Historical Review.  

7. Relations: The Second Time Around.
 

Relations Revisited: Properties of Relations.  

 

Computer Recognition: Zero-One Matrices and Directed Graphs.  

 

Partial Orders: Hasse Diagrams.  

 

Equivalence Relations and Partitions.  

 

Finite State Machines: The Minimization Process.  

 

Summary and Historical Review.  

PART 2. FURTHER TOPICS IN ENUMERATION.

8. The Principle of Inclusion and Exclusion.
 

The Principle of Inclusion and Exclusion.  

 

Generalizations of the Principle.  

 

Derangements: Nothing Is in Its Right Place.  

 

Rook Polynomials.  

 

Arrangements with Forbidden Positions.  

 

Summary and Historical Review.  

9. Generating Functions.
 

Introductory Examples.  

 

Definition and Examples: Calculational Techniques.  

 

Partitions of Integers.  

 

The Exponential Generating Functions.  

 

The Summation Operator.  

 

Summary and Historical Review.  

10. Recurrence Relations.