Discrete Mathematical Structures

1. Fundamentals

1.1 Sets and Subsets

1.2 Operations on Sets

1.3 Sequences

1.4 Properties of Integers

1.5 Matrices

1.6 Mathematical Structures

 

2. Logic

2.1 Propositions and Logical Operations

2.2 Conditional Statements

2.3 Methods of Proof

2.4 Mathematical Induction

2.5 Mathematical Statements

2.6 Logic and Problem Solving

 

3. Counting

3.1 Permutations

3.2 Combinations

3.3 Pigeonhole Principle

3.4 Elements of Probability

3.5 Recurrence Relations 112

 

4. Relations and Digraphs

4.1 Product Sets and Partitions

4.2 Relations and Digraphs

4.3 Paths in Relations and Digraphs

4.4 Properties of Relations

4.5 Equivalence Relations

4.6 Data Structures for Relations and Digraphs

4.7 Operations on Relations

4.8 Transitive Closure and Warshall's Algorithm

 

5. Functions

5.1 Functions

5.2 Functions for Computer Science

5.3 Growth of Functions

5.4 Permutation Functions

 

6. Order Relations and Structures

6.1 Partially Ordered Sets

6.2 Extremal Elements of Partially Ordered Sets

6.3 Lattices

6.4 Finite Boolean Algebras

6.5 Functions on Boolean Algebras

6.6 Circuit Design

 

7. Trees

7.1 Trees

7.2 Labeled Trees

7.3 Tree Searching

7.4 Undirected Trees

7.5 Minimal Spanning Trees

 

8. Topics in Graph Theory

8.1 Graphs

8.2 Euler Paths and Circuits

8.3 Hamiltonian Paths and Circuits

8.4 Transport Networks

8.5 Matching Problems

8.6 Coloring Graphs

 

9. Semigroups and Groups

9.1 Binary Operations Revisited

9.2 Semigroups

9.3 Products and Quotients of Semigroups

9.4 Groups

9.5 Products and Quotients of Groups

9.6 Other Mathematical Structures

  

10. Groups and Coding

10.1 Coding of Binary Information and Error Detection

10.2 Decoding and Error Correction

10.3 Public Key Cryptology

 

11. Languages and Finite-State Machines

11.1 Languages

11.2 Representations of Special Grammars and Languages

11.3 Finite-State Machines

11.4 Monoids, Machines, and Languages

11.5 Machines and Regular Languages

11.6 Simplification of Machines

 

Appendix A: Algorithms and Pseudocode

Appendix B: Additional Experiments in Discrete Mathematics

Appendix C: Coding Exercises