Based on the Text “ Discrete Mathematics & Its Applications” (5 th Edition) by Kenneth H. Rosen

1. Tzu-Chi UniversityDept. of Medical Informatics MI112A & MI112BDiscrete MathematicsJen-Liang Cheng Ph.D.(鄭仁亮) Based on the Text “Discrete Mathematics & Its Applications” (5th Edition)by Kenneth H. Rosen (c)2001-2003, Michael P. Frank

2. Module #0:Course Overview (c)2001-2003, Michael P. Frank

3. What is Mathematics, really? • It’s not just about numbers! • But, the concepts can relate to numbers, symbols, visual patterns, or anything! Mathematics is the study of any and all <absolutely certain truths> about any and all perfectly well-defined concepts. (c)2001-2003, Michael P. Frank

4. So, what’s this class about? What are “discrete structures” anyway? • “Discrete” - Composed of distinct, separable parts. (Opposite of continuous.)discrete:continuous :: digital:analog • “Structures” - objects built up from simpler objects according to a definite pattern. • “Discrete Mathematics” - The study of discrete, mathematical objects and structures. (c)2001-2003, Michael P. Frank

5. Propositions Predicates (Discrete) Functions Orders of Growth Algorithms Integers Boolean Algebra Finite-State Machine Proofs Summations Permutations Combinations Relations Graphs Trees Discrete Structures We’ll Study (c)2001-2003, Michael P. Frank

6. Relationships Between Structures • “→” :≡ “Can be defined in terms of” Programs Proofs Groups Trees Operators Propositions Complex numbers Graphs Real numbers Strings Functions Integers Naturalnumbers Matrices Relations Sequences Infiniteordinals Bits n-tuples Vectors Sets (c)2001-2003, Michael P. Frank

7. Some Notations We’ll Learn (c)2001-2003, Michael P. Frank

8. Why Study Discrete Math? • The basis of all of digital information processing: Discrete manipulations of discrete structures represented in memory. • It’s the basic language and conceptual foundation of all of computer science. • Discrete concepts are also widely used throughout math, science, engineering, economics, biology, etc., … • A generally useful tool for rational thought! (c)2001-2003, Michael P. Frank

9. Advanced algorithms & data structures Programming language compilers & interpreters. Computer networks Operating systems Computer architecture Database management systems Cryptography Error correction codes Graphics & animation algorithms, game engines Just about everything! Uses for Discrete Math in Computer Science (c)2001-2003, Michael P. Frank

10. Logic (§§1.1-1.4) Proof methods (§1.5) Functions (§1.8) Proof strategy (§3.1) Sequences (§3.2) Summations (§3.2) Inductive proofs (§3.3) Algorithms (§2.1) Orders of Growth (§2.2) Complexity (§2.3) Matrices (§2.7) Counting (ch. 4) Recurrences (§6.1-6.3) Relations (ch. 7) Graph Theory (chs. 8+9) Boolean Algebra Finite State Machine Course Outline (as per Rosen) (c)2001-2003, Michael P. Frank

11. Course Objectives • Upon completion of this course, the student should be able to: • Check the validity of simple logical arguments. • Check the correctness of simple algorithms. • Creatively construct simple valid logical arguments. • Creatively construct simple correct algorithms. • Describe the definitions and properties of a variety of specific types discrete structures. • Correctly read, write and analyze various types of structures using standard notations. Think! (c)2001-2003, Michael P. Frank