이산수학 (Discrete Mathematics) 7.3 관계의 표현 (Representing Relations)

이산수학 (Discrete Mathematics) 7.3 관계의 표현 (Representing Relations) 2006년 봄학기 문양세 강원대학교 컴퓨터과학과

Representing Relations 7.3 Representing Relations • Some ways to represent n-ary relations: • With an explicit list or table of its tuples. • With a function,or with an algorithm for computing this function. • Some special ways to represent binary relations: • With a zero-one matrix. • With a directed graph.

Using Zero-One Matrices 7.3 Representing Relations To represent a relation R by a matrix MR = [mij], let mij = 1 if (ai,bj)R, else 0. E.g., Joe likes Susan and Mary, Fred likes Mary, and Mark likes Sally. The 0-1 matrix representationof that “Likes”relation:

Zero-One Reflexive, Symmetric (1/2) 7.3 Representing Relations • Terms: Reflexive, irreflexive, symmetric, and antisymmetric. • These relation characteristics are very easy to recognize by inspection of the zero-one matrix. any-thing any-thing any-thing any-thing Irreflexive:all 0’s on diagonal Reflexive:all 1’s on diagonal

Zero-One Reflexive, Symmetric (2/2) 7.3 Representing Relations anything anything Symmetric:all identicalacross diagonal Antisymmetric:all 1’s are acrossfrom 0’s

Using Directed Graphs (1/2) 7.3 Representing Relations A directed graph or digraphG=(VG,EG) is a set VGof vertices (nodes) with a set EGVG×VG of edges (arcs,links).(관계는 노드(꼭지점)의 집합 V와 에지(링크)의 집합 E로 표현되는 방향성 그래프로 나타낼 수 있다.) Visually represented using dots for nodes, and arrows for edges. Notice that a relation R:A↔B can be represented as a graph GR=(VG=AB, EG=R).(일반적으로, 노드는 점으로, 에지는 화살표로 표현한다.)

Using Directed Graphs (2/2) 7.3 Representing Relations Edge set EG(blue arrows) GR MR Joe Susan Fred Mary Mark Sally Node set VG(black dots)

Relational Databases (관계형 DB) 7.2 n-ary Relations A relational database is essentially an n-ary relation R.(관계형 데이터베이스란 n-항 관계 R을 의미한다.) A domain Ai is a primary key for the database if the relation R contains at most one n-tuple (…, ai, …) for any value ai within Ai.(만일 R이 (정의역 Ai에 포함된) ai에 대해서 기껏해야 하나의 n-항 튜플 (…, ai, …)를 포함하면, Ai는 기본 키라 한다.)(다시 말해서, ai값을 가지는 n-항 튜플이 유일하면 Ai를 키본 키라 한다.) A composite key for the database is a set of domains {Ai, Aj, …} such that R contains at most 1 n-tuple (…,ai,…,aj,…) for each composite value (ai, aj,…)Ai×Aj×…

Digraph Reflexive, Symmetric 7.3 Representing Relations It is extremely easy to recognize the reflexive/irreflexive/ symmetric/antisymmetric properties by graph inspection.            Reflexive:Every nodehas a self-loop Irreflexive:No nodelinks to itself Symmetric:Every link isbidirectional Antisymmetric:No link isbidirectional Asymmetric, non-antisymmetric Non-reflexive, non-irreflexive

Homework #8 7.3 Representing Relations $7.1의 연습문제: 4(b, d), 24 $7.2의 연습문제: 2, 6 $7.3의 연습문제: 2(b,d), 19(b,d) Due Date:

이산수학 (Discrete Mathematics) 7.3 관계의 표현 (Representing Relations)