# Discrete Mathematics and Exercise

Lecturer Keiichirou KUSAKARI, Associate Professor School of Engineering / Graduate School of Engineering, 2013 Spring Electrical and Electronic Engineering and Information Engineering freshman (3・1 session / week 15 weeks / semester)

### Course Aims

In this lecture, we will be learning about the basic concepts of set, function and relation. We will also focus on the elementary number theory and basic theories of algebra. In the exercise session you will be asked to establish proofs for exercise problems to get the basic understanding of the format.

### Key Features

Lecture notes are uploaded for your convenience. Therefore, in this lecture, I would like you to focus on the lecture not just on taking notes. It is true that writing equations yourself by taking notes is important to get the fundamental understanding of the lecture. However, in this lecture students will be asked to solve lots of problems to get the fundamental techniques of writing logically well-constructed proofs.

Mid term exam will be in the last lecture. Problems on the final exam will be the same one as in the mid term exam. It is because student will be forced to review all the problems on the mid term exam and this is good way for them to review and get them a chance to study.

### Contents

• Basic concepts about set, function and relation

• Elementary number theory

• Basic theories of algebra

### Course Overview

Objectives of our course are as followed.

• Getting sufficient knowledge of basic mathematics in order to get through four-year study in Nagoya University

• Getting sufficient knowledge of mathematical notational conventions in order to get through four-year study in Nagoya University

• Obtaining logical thinking, or the ability to prove logically and clearly without any unclear reasoning. (Many example problems will be given in the exercise session) Logically well-constructed proof has to follow following rules.

• No inconsistency in the proof

• No superfluous discussion

• Enough reasoning for the conclusion.

• Clear storyline

• Development of abstract thinking

This lecture is designed for Electrical and Electronic Engineering and Information Engineering freshman. Therefore, one of our objectives is to get the basic skills and knowledge of basic mathematics. Furthermore, through intensive exercise I would like you to obtain logical thinking for writing thesis in senior year.

### Assignments

Followings are the example problems given in the exercise session.

### Course Schedule

Session Contents
1 Set, Proposition and Relation (1)
2 Set, Proposition and Relation (2)
3 Relation and Order (1)
4 Relation and Order (2)
5 Enumerative combinatorics
6 Divisor, Multiple and Euclidean algorithm
7 Prime Number (1)
8 First Order Diophantine Equation
9 Congruence expression and Chinese remainder theorem
10 Prime Number (2)
11 Group
12 Ring, Integral Domain, and Field
13 Polynomial Rings and Galois Field
14 Mid term exam
15 Summary / Final term exam

### Lecture Handouts

Note: All files are in Japanese.

Session #1 and #2

Session #3 and #4

Session #5

Session #6

Session #7

Session #8

Session #9

Session #10

Session #11

Session #12

Session #13