Interests  in computer science and the use of computer applications, together with connections to many real-world situations, have helped make topics of discrete mathematics more commonplace  in  school and university curricula.   A topic of widespread application and interest is combinatorics,  the study of counting techniques.  Enumeration, or counting, may strike one as an obvious process that a student learns when first studying arithmetic. But then, it seems, very little attention is paid to further developments in counting as the student turns to "more difficult" areas in mathematics, such as algebra, geometry, trigonometry, and calculus. . . . Enumeration [however] does not end with arithmetic. It also has applications in such areas as coding theory, probability, and statistics (in mathematics) and in the analysis of algorithms (in computer science). [Ralph P. Grimaldi, in Discrete and Combinatorial Mathematics, 1994, p. 3]   

  Combinatorial Analysis is an area of mathematics concerned with solving problems for which the number of possibilities is finite (though possibly quite large). These problems may be broken into three main categories: determining existence, counting, and optimization. Sometimes it is not clear whether a problem has a solution or not. This is a question of existence. In other cases solutions are known to exist, but we want to know how many there are. This is a counting problem. Or a solution may be desired that is "best" in some sense. This is an optimization problem. [John A. Dossey, Albert D. Otto, Lawrence E. Spence, & Charles V. Eynden, in Discrete Mathematics, 1987, p. 1]

   Current documents that support the reform of school mathematics education suggest the need for increased attention to topics in discrete mathematics as well as in probability and statistics. The topic of combinatorics--counting--is mentioned in the National Council of Teachers of Mathematics standards documents and in the Mathematical Association of America's recommendations for teacher preparation as a topic area worthy of study by middle school and high school teachers.

   This quarter, we will study and apply combinatorial techniques in a variety of settings.  In doing so, we will make connections to algebra, probability, and many other topics in mathematics. During the course, we may also study the process of proof by induction, the use of recursion, and the graph theory and knot theory and more.

Room 82-105,  충북대학교 사범대학
Wednesday  3:00-5:00 pm
E-mail: wkkim@chungbuk.ac.kr
FAX:  043-275-2715

Problem Sets & Quizzes (20%)
These will be weekly assignments by webboard,  and several quizzes during the course.

Test  (20%)
One exam will be given during the course.
The test is tentatively scheduled for the mid week of the Quarter.

Final Examination (60%)

The test is scheduled for the last week of the Quarter.

 1 주

이산수학에서 전형적인 네 가지 문제

1장 1절 MAGIC CARD문제

1장 2절 비둘기집의 원리

1장 3절 하노이탑 문제

1장 4절 격자다각형 문제

 2 주

다양한 세기 방법

2장 1절 두가지 세기(counting)의 방법

2장 2절 순서가 없는 선택의 문제

2장 3절  확률(probability)과 그 응용

2장 4절  조건부 확률과 그  응용  

 3 주

이항정리와 그 응용

3장 1절  이항정리

3장 2절  다양한 이항항등식

3장 3절  다항정리

3장 4절  다항정리와 그 응용

 4 주

Enumeration

4장 1절  Fibonacci 수열과 그 응용

4장 2절  선형 Diophantine 방정식

4장 3절  포함배제의 원리

4장 4절  포함배제의 원리의 응용

 5 주

More Enumeration

5장 1절  Pascal의 삼각형의 응용   

5장 2절  분할 

5장 3절  Generating functions 

5장 4절  Golden ratio and Fibonacci NumberI

6 주

recursion과  

그 응용 I

6장 1절  Fibonacci 수열과 점화식  I

6장 2절  Fibonacci 수열과  점화식  I I  

6장 3절  수학적 귀납법과 유한차분

6장 4절  포함배제의 원리 I I  

7 주

recursion과  

그 응용 II

7장 1절  Recursion과  Sierpinski Gasket  

7장 2절  Recursion과  Zeno's Paradox

7장 3절  Recursion and Fractal

7장 4절  Recursion and Chaos

8 주

그래프이론

8장 1절  그래프의 개념

8장 2절  subgraphI

8장 3절  regular & bipartite graph  

8장 4절  Eulerian & Hamiltonian graphI

9 주

여러 가지 discrete한

구조들

9장 1절  여러 가지 함수 개념 

9장 2절  Relations and Digraphs

9장 3절  Order Relations and Lattices

9장 4절 Iterated Function

10주

암호이론과 그 응용 

10장 1절  Coding Theory

10장 2절  An Error-Correcting Code

10장 3절 Hamming 1-Error Correcting Code

10장 4절 Hamming 1-Error Correcting Code

11주

Four Color Problem and Knot Theory 

11장 1절  Semigroups and Groups

11장 2절  Finite State Machines

11장 3절  Four Color Problem

11장 4절  Knot  Theory

12주

Final Examination