1. Discrete Math II - Introduction -
6/25/2018
Discrete Math II - Introduction -
Howon Kim

2. 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

3. 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)
Phone:
Homepage :
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 !

4. 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

5. 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

6. 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)

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

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

14. 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

15. 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

16. 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

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

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

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