Discrete Mathematics Relations and Its Properties 2013 Binary

이산수학(Discrete Mathematics) 관계와 그 특성 (Relations and Its Properties) 2013년 봄학기 강원대학교 컴퓨터과학전공 문양세

Binary Relations (이진 관계) 7. 1 Relations & Its Properties Let A, B be any two sets. A binary relation R from A to B, written R: A↔B, is a subset of A×B. (A에서 B로의 이진 관계 R은 R: A↔B로 표기하며 A×B의 부분집합이다. ) • E. g. , let < : N↔N : ≡ {(n, m) | n < m} The notation a R b or a. Rb means (a, b) R. • E. g. , a < b means (a, b) < If a. Rb, we may say “a is related to b (by relation R). ” (a. Rb이면, “a는 (관계 R에 의해서) b에 관계된다”고 말한다. ) Page 2 Discrete Mathematics by Yang-Sae Moon

Complementary Relations (보수 관계) 7. 1 Relations & Its Properties Let R: A↔B be any binary relation. Then, R: A↔B, the complement of R, is the binary relation defined by R : ≡ {(a, b) | (a, b) R} = (A×B) − R Note the complement of R is R. Example: < = {(a, b) | (a, b) <} = {(a, b) | ¬(a<b)} = ≥ Page 3 Discrete Mathematics by Yang-Sae Moon

Complementary Relation Example 7. 1 Relations & Its Properties 예제: A = {0, 1, 2}, B = {a, b}라 하면, {(0, a), (0, b), (1, a), (2, b)}는 A에서 B로 의 관계 R로 표현할 수 있다. 이 때, • (0, a) R 이므로, 0 Ra라 할 수 있다. • 그러나, (1, b) R 이므로, 1 Rb라 할 수 있다. Page 4 Discrete Mathematics by Yang-Sae Moon

Inverse Relations (역 관계) 7. 1 Relations & Its Properties Any binary relation R: A↔B has an inverse relation R− 1: B↔A, defined by R− 1 : ≡ {(b, a) | (a, b) R}. E. g. , if R: People↔Foods is defined by a. Rb a eats b, then: b R− 1 a b is eaten by a. (Passive voice. ) (R− 1 will be “is eaten by. ”) Page 5 Discrete Mathematics by Yang-Sae Moon

Relations on a Set 7. 1 Relations & Its Properties A (binary) relation from a set A to itself is called a relation on the set A. (집합 A에서 A로의 관계를 집합 A상의 관계라 한다. ) E. g. , the “<” relation from earlier was defined as a relation on the set N of natural numbers. (“<”은 정수 집합 N에 대한 관계이다. ) The identity relation IA on a set A is the set {(a, a)|a A}. (집합 A에 대한 항등 관계 IA는 집합 {(a, a)|a A}를 의미한다. ) Page 6 Discrete Mathematics by Yang-Sae Moon

Examples of Relations on a Set (1/2) 7. 1 Relations & Its Properties 예제: A = {1, 2, 3, 4}라 할 때, 관계 R = {(a, b)| a divides b}에 속하는 순서쌍은 ? • A x A의 원소인 (a, b)에 있어서 b를 a로 나눌 수 있는 순서쌍을 구한다. • 즉, R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}이다. Page 7 Discrete Mathematics by Yang-Sae Moon

Reflexivity (반사성) 7. 1 Relations & Its Properties A relation R on A is reflexive if a A, a. Ra. • E. g. , the relation ≥ : ≡ {(a, b) | a≥b} is reflexive. • 즉, (a, a)를 원소로 가지면 반사적(reflexive)이라고 이야기한다. A relation is irreflexive iff its complementary relation is reflexive. (역관계가 반사이면, 해당 관계는 비반사이다) • Example: < is irreflexive. Page 9 Discrete Mathematics by Yang-Sae Moon

Symmetry & Antisymmetry (대칭성) 7. 1 Relations & Its Properties A binary relation R on A is symmetric iff R = R− 1, that is, if (a, b) R ↔ (b, a) R. • E. g. , = (equality) is symmetric. < is not. • “is married to” is symmetric, but “likes” is not. • 즉, (a, b)가 R의 원소일 때, 반드시 (b, a)도 원소이면 대칭적이라 한다. A binary relation R is antisymmetric if (a, b) R → (b, a) R. • < is antisymmetric, “likes” is also antisymmetric. Page 11 Discrete Mathematics by Yang-Sae Moon

Symmetry & Antisymmetry Example 7. 1 Relations & Its Properties 예제: 양의 정수 집합에 대한 “나누다” 관계는 대칭인가? 반대칭인가? • 반례(counterexample)를 들어 반대칭임을 보인다. • 즉, 1|2 이지만 2|1이므로, 반대칭이다. Page 12 Discrete Mathematics by Yang-Sae Moon

Transitivity (전이성) 7. 1 Relations & Its Properties A relation R is transitive iff (for all a, b, c) (a, b) R (b, c) R → (a, c) R. A relation is intransitive if it is not transitive. Examples: “is an ancestor of” is transitive. “likes” is intransitive. Page 13 Discrete Mathematics by Yang-Sae Moon

Composite Relations (관계 합성/결합) 7. 1 Relations & Its Properties Let R: A↔B, and S: B↔C. Then the composite S R of R and S is defined as: S R = {(a, c) | b: a. Rb b. Sc} ((a, b) R이고 (b, c) S이면, S R은 (a, c)을 원소로 하는 관계이다. ) Note function composition f g is an example. The nth power Rn of a relation R on a set A (A에 대한 관계 R의 n 제곱) can be defined recursively by: R 0 : ≡ IA ; Rn+1 : ≡ Rn R Page 15 for all n≥ 0. Discrete Mathematics by Yang-Sae Moon

Examples of Composite Relations (2/2) 7. 1 Relations & Its Properties 예제: R = {(1, 1), (2, 1), (3, 2), (4, 3)}이라 하자. n = 2, 3, 4, … 일 때, 거듭 제곱 Rn을 구하라. • R 2 = R R = {(1, 1), (2, 1), (3, 1), (4, 2)} • R 3 = R 2 R = {(1, 1), (2, 1), (3, 1), (4, 1)} • R 4 = R 3 R = {(1, 1), (2, 1), (3, 1), (4, 1)} • … • Rn = Rn-1 R = {(1, 1), (2, 1), (3, 1), (4, 1)} You can get Rn using “induction. ” (교재의 R 3와 R 4는 잘못 구해진 것으로 보임…) Page 17 Discrete Mathematics by Yang-Sae Moon