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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

이산수학 (Discrete Mathematics) n- 항 관계 (n- ary Relations). 2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세. n-ary Relations (n- 항 관계 ). n- ary Relations.

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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

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  1. 이산수학(Discrete Mathematics) n-항 관계 (n-ary Relations) 2014년 봄학기 강원대학교 컴퓨터과학전공 문양세

  2. n-ary Relations (n-항 관계) n-aryRelations An n-ary relation R on sets A1,…,An, written R:A1,…,An, is a subset R  A1× … × An.(A1,…,An에 대한 n-항 관계 R은 A1× … × An의 부분집합이다.) The sets Ai are called the domains of R.(Ai를 R의 정의역이라 한다.) The degree of R is n. (관계 R의 차수는 n이다.)

  3. Relational Databases (관계형 DB) n-aryRelations 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×…

  4. Primary Key 예제 n-aryRelations 예제: (새로운 튜플이 추가되지 않는다고 할 때,) 다음 테이블에서 어떤 정의역이 기본 키인가? • Student_name은 키본 키이다. (유일하게 구분 짓는다.) • 마찬가지로, ID_number 또한 기본 키이다. • 반면에, Major나 GPA는 기본 키가 아니다.

  5. Composite Key 예제 n-aryRelations 예제: (새로운 튜플이 추가되지 않는다고 할 때,) 다음 테이블에서 {Major, GPA}는 합성 키인가? • Major와 GPA를 조합하여 사용하면 튜플을 유일하게 구분 지을 수 있으므로, {Major, GPA}는 상기 테이블의 합성 키이다.

  6. Selection Operator () n-aryRelations • Let A be any n-ary domain A=A1×…×An, and let P:A→{T,F} be any predicate on elements of A.(A를 n-항 관계의 정의역이라 하고, P를 A에서 {T,F}로의 술어라 하자.) • Then, the selection operatorp is the operator that maps any (n-ary) relation R on A to the n-ary relation of all n-tuples from R that satisfy P.(셀렉션 연산자 p은 관계 R의 n-튜플 중에서 술어 P를 만족하는 튜플들의 관계로 정의한다.) • I.e., RA,p(R) = R{aA | P(a) = T}

  7. Selection Example n-aryRelations Suppose we have a domain A = StudentName × Level × SocSecNos Suppose we define a certain predicate on A, UpperLevel(name, level, ssn) :≡ [(level = junior) (level = senior)] Then, UpperLevel is the selection operator that takes any relation R on A (database of students) and produces a relation consisting of just the upper-level classes (juniors and seniors). That is, (level = junior)  (level = senior)(R)

  8. Projection Operator () n-aryRelations • Let A = A1×…×An be any n-ary domain, and let {ik}=(i1,…,im) be a sequence of indexes, • That is, where 1 ≤ ik ≤ n for all 1 ≤ k ≤ m. • Then the projection operator on n-tuplesis defined by:

  9. Projection Example n-aryRelations Suppose we have a ternary (3-ary) domainCars = Model×Year×Color. (note n=3). Consider the index sequence {ik}= 1,3. (m=2) Then the projection simply maps each tuple (a1,a2,a3) = (model,year,color) to its image:

  10. (Natural) Join Operator () n-aryRelations • Puts two relations together to form a sort of combined relation. (관계를 합성하는 한 가지 방법) • If the tuple (A,B) appears in R1, and the tuple (B,C) appears in R2, then the tuple (A,B,C) appears in the joinR1R2. • A, B, C can also be sequences of elements rather than single elements.

  11. (Natural) Join Example n-aryRelations Suppose R1 is a teaching assignment table, relating Professors to Courses. ((Professor, Courses)로 구성된 관계) Suppose R2 is a room assignment table relating Courses to Rooms,Times. ((Courses, Rooms, Times)로 구성된 관계) Then R1R2 is like your class schedule, listing (professor,course,room,time).

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