Discrete Mathematics 이재원 School of Information Technology Sungshin W. University (Special Thanks to the original author of this lecture notes, Prof. 홍기형) 2008-fall

Text and References Text Discrete Mathematics and Its Applications, 6th Edition, Kenneth H. Rosen 2008-fall

Professor Info 이재원 질문 및 상담 Course Home jwlee@sungshin.ac.kr 수정관 811호 수업시간 전후 E-mail로 신청 Course Home http://cs.sungshin.ac.kr/~jwlee/dm2/dm.htm 2008-fall

Grading Mid : 35% Final Exam : 35% Home Work : 10% 출석 : 20% Handwriting : word processor 하지 말 것. 출석 : 20% 2008-fall

What is Mathematics, really? It’s not just about numbers! Mathematics is much more than that: But, the concepts can relate to numbers, symbols, visual patterns, or anything! Mathematics is, most generally, the study of any and all absolutely certain truths about any and all perfectly well-defined concepts. As you may have already concluded from the title of the textbook, this is a course about mathematics. But, it is a type of mathematics that may be unfamiliar to you. You may have concluded from your exposure to math so far that mathematics is primarily about numbers. But really, that’s not true at all. The essence of mathematics is just the study of formal systems in general. A formal system is any kind of entity that is defined perfectly precisely, so that there can be no misunderstanding about what is meant. Once such a system has been set forth, and a set of rules for reasoning about it has been laid out, then, anything you can deduce about that system while following those rules is an absolute certainty, within the context of that system and those rules. Number systems aren’t the only kinds of system that we can reason about, just one kind that is widely useful. For example, we also reason about geometrical figures in geometry, surfaces and other spaces in topology, images in image algebra, and so forth. Any time you precisely define what you are talking about, set forth a set of definite axioms and rules for some system, then it is by definition a mathematical system, and any time you discover some unarguable consequence of those rules, you are doing mathematics. In this course, we’ll encounter many kinds of perfectly precisely defined (and thus, by definition, mathematical) concepts other than just numbers. You will learn something of the richness and variety of mathematics. And, we will also teach you the language and methods of logic, which allow us to describe mathematical proofs, which are linguistic presentations that establish conclusively the absolute truth of certain statements (theorems) about these concepts. In a sense, it’s easy to create your own new branch of mathematics, never before discovered: just write down a set of axioms and inference rules, make sure their meanings are perfectly clear, and then work out their consequences. Whether anyone else finds your new area of mathematics particularly interesting is another matter. The areas of math that have been thoroughly explored by large numbers of people are generally ones that have a large number of applications in helping to understand some other subject, whether it be another area of mathematics, or physics, or economics, biology, sociology, or whatever other field. See http://mathworld.wolfram.com/ for a survey of some other areas of mathematics, to show you just how much there is to mathematics, besides just numbers!

So, what’s this class about? What are “discrete structures” anyway? “Discrete” ( “discreet”!) - Composed of distinct, seperable 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.

Discrete Structures We’ll Study Propositions Predicates Sets (Discrete) Functions Orders of Growth Algorithms Integers Proofs Summations Permutations Combinations Relations Graphs Trees Boolean Algebra Logic Circuits Here are just a few of the kinds of discrete structures and related concepts that we’ll be studying this semester.

Relationships Between Structures “→” :≡ “Can be defined in terms of” Programs Groups Proofs Trees Operators Propositions Complex numbers Graphs Real numbers Strings Functions Integers Natural numbers Matrices Relations As I mentioned earlier, mathematical structures can often be built up or defined in terms of simpler structures. This diagram (which itself is an example of the type of discrete structure known as a “graph”) outlines some ways in which a variety of discrete (and continuous) structures can ultimately be built up from a very simple type of discrete structure known as a set, which is one of the first structures we will learn. This is just to give you a a bit of a feel for the inter-relatedness of mathematical objects. This diagram is vastly simplified; some of the dependencies are left out, many other kinds of structures and ways of defining structures in terms of other structures are also possible and are left out for simplicity. For example, sets themselves can be defined in terms of functions, or relations. No one type of structure is truly fundamental, since almost any structure can be defined in terms of almost any other. Sets are only one possible starting point; but they are a popular one because their definition is so simple. We will revisit this diagram in more detail throughout this course, and show how the various links work. Sequences Infinite ordinals Bits n-tuples Vectors Sets

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!

Uses for Discrete Math in Computer Science 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! SSWU MIPS Lab. Ki-Hyung Hong

Course Outline (as per Rosen) Logic (§1.1-1.4) Proof methods (§1.5-1.7) Set theory (§2.1-2.2) Functions (§2.3) Sequences & Summations (§2.4) Algorithms (§3.1) Orders of Growth & Complexity (§3.2-3.3) Number Theory (§3.4-3.8) Recursion(§4.1-4.4) Counting (§5.1-5.3) Discrete Probability (§6.1) Recurrence Relations (§7.1, 7.3) Relations (§8.1-8.6) Graph (Tree) Theory (§9.1-9.5, 10.1-10.3) Boolean Algebra (§11.1-11.4) Here is the order of topics covered, and corresponding section numbers in the textbook (5th edition). We are basically following the same order of material as the textbook. However, we will skip over some sections in the later chapters.

Symbol, Notation (기호, 표기법) 1 What is this ? Think: your name, symbol ‘+’, ‘seoul’, … 2+2 ? “Seoul” + “, Korea” ?

Basics for studying Math. and Languages Understanding the semantics of Symbols and Notations 1, 2, 3, …, 9, 0 Positional semantics : 912 vs. 219 Both are consisting the same 3 symbols, but are different. x, y (variables) Each of them can have a value from a designated domain. x + y = 10, true when x=4, y=6. false when x=5, y=6. x + y < x * y, true for positive integers x and y I am a student. Iamastudent. for (i=1; i<10; i++) x=x+1;

Some Notations We’ll Learn