Discrete Math II - Introduction - 6/25/2018 Discrete Math II - Introduction - Howon Kim 2017.9.4

About this course… Course name : Discrete Math II (CP21697) Study the basics on number theory, graph theory, automata, and mathematical techniques for computer science & engineering Number theory (finite fields and ring) is the fundamental knowledge for the cryptography & security and Coding Theory etc. Graph theory & automata is the basic mathematical techniques to understand the computer science, networks and many topics in computer engineering

About this course… About Instructor Major Research Interests 6/25/2018 About Instructor Office : A06-503 Office hours : 12:30 ~ 13:30 PM(Monday, Wednesday) Email: howonkim@gmail.com, howonkim@pusan.ac.kr Phone: 010-8540-6336 Homepage : http://infosec.pusan.ac.kr Major Research Interests 사물인터넷 연구센터 IoT(Internet of Things: 지능형 사물 네트워크) 기술 연구 머신러닝/딥러닝 기술 연구 정보보호/해킹, 네트워크 보안, 암호 기술, IoT 보안 연구 FPGA & ASIC chip design Recruit Ambitious Students !

About this course… Textbook Time & Classroom References 6/25/2018 Textbook “Discrete and combinatorial mathematics ” (5th Ed), R.P. Grimaldi, 2004 Selected Materials for mathematical techniques Time & Classroom 15:00 ~ 16:15 PM (Monday, Wednesday), A6-202 References Discrete mathematics by Richard Johnsonbaugh Introduction to Automata Theory, Languages, and Computation by John E. Hopcroft Introduction to Graph Theory by Douglas B West A Course in Number Theory and Cryptography by Neal Koblitz

About this course 활동사항 학습내용 주 Algebra 개요 Groups, Integers modulo N 6/25/2018 활동사항 학습내용 주 Algebra 개요 1 Groups, Integers modulo N 2 Ring 3 Field 4 Applications of number theory 5 Midterm exam 1 6 Introduction 7 Euler Circuit 8 Planar Graph, Hamiltonian Path, Coloring 9 Midterm exam 2 10 Set theory of strings 11 Regular languages, finite automata 12 Grammars and languages 13 Theory of computation 14 Final Exam 15 16

About this course Grading Policy 점 수 항 목 15 Attendance 25/25 6/25/2018 Grading Policy 점 수 항 목 15 Attendance 25/25 Midterm exam 1/2 25 Final 10 Homework 100 Total (수업시간 참여 충실도 반영)

Algebra Definition K : a set of data Operator opj Tuple <K, op1, op2, …, opn> < R, , , ,  > < {T,F }, , ,  > ; Boolean algebra K : a set of data |K| : order finite or infinite Operator opj Closure opj : Ki  K Unary if i=1, Binary if i=2, … 7

Identity and Zero  : K  K  K Identity element e for  in K (항등원) e  a = a  e = a for all a ∈ K Zero element z for  in K (영원) z  a = a  z = z for all a ∈ K Examples < Z, + > Identity : 0, Zero : none < Z,  > Identity : 1, Zero : 0 8

Inverse  : K  K  K Let e be the identity element for  in K. Left inverse a’L  a = e , a ∈ K Right inverse a  a’R = e , a ∈ K If a’L = a’R = a’ , a’ is the inverse of a. Example < Z, + > Identity 0, (-x) is the inverse of x : x + (-x) = (-x) + x = 0 9

Properties of Operator Let  : K  K  K be a binary operator. (1) Closure (2) Associative (a  b)  c = a  (b  c) for all a, b, c ∈ K. (3) Identity There is an identity element e ∈ K for . (4) Inverse For each a ∈ K, there is an inverse a’ ∈ K for . (5) Commutative a  b = b  a for all a,b ∈K. 10

Binary Algebra < K,  > for binary operator  : K  K  K Semigroup (반군) : Associative < Z+, + > A semigroup is a set with an associative binary operation which satisfies closure and associative law. Monoid (단위반군) : Associative, Identity < N, + >, < Z,  >, < {T,F },  > A monoid is a set that is closed under an associative binary operation and has an identity element Group (군) : Associative, Identity, Inverse < Z, + > Abelian group (대수군) : Associative, Identity, Inverse, Commutative 11

Binary Algebra Properties < K,  > Closure Associative Identity Set (1), (2) Semigroup Properties Closure Associative Identity Inverse Commutative (5) Abelian Semigroup Monoid (3) Abelian Monoid (5) Group (4) Abelian Group (5) 12

Binary Algebra Set Closure Semigroup Associative Commutative Monoid Abelian Group Abelian Monoid Abelian Semigroup Monoid Identity Group Inverse 13

Ring ( Two operators ) < K, , > Conditions for Ring Two binary operators ,  : K  K  K Conditions for Ring < K, > is an abelian group.  is associative  is distributive over  a  (b  c) = (a  b)  (a  c) and (a  b)  c = (a  c)  (b  c) for all a,b,c ∈ K. 14

Definitions < K, , > Conditions for operator  : < K, > : abelian group, and distribution laws hold Conditions for operator  : Ring (환) : Associative Ring with Unity : Associative, Identity Commutative Ring : Associative, Commutative Commutative Ring with Unity Associative, Identity, Commutative Field (체) Associative, Identity, Commutative, Inverse 15

Ring and Field Properties for  < K, , > (0) Distributive Set (0), (1), (2) Ring Properties for  (0) Distributive (1) Closure (2) Associative (3) Identity (4) Inverse (5) Commutative (5) Commutative Ring (3) Ring with Unity Commutative Ring with Unity (5) (3) Field (4) 16

Ring and Field < K, , > Closure Distributive Ring Associative Ring with Unity Identity Commutative Ring Commutative Field Inverse Commutative Ring with Unity 17

Next… 6/25/2018 Basics on Number Theory…

Q&A Thank you for your attention. Thank you. 19