2. 2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세 이산수학 (Discrete Mathematics)  집합 연산 (Set Operations)

3. Discrete Mathematics by Yang-Sae Moon Page 2 Union Operator ( 합집합 ) For sets A, B, their union A  B is the set containing all elements that are either in A, or (“  ”) in B (or, of course, in both). (A 또는 B 에 속하거나 양쪽에 모두 속하는 원소들의 집합 ) Formally, A  B = {x | x  A  x  B}. Note that A  B contains all the elements of A and it contains all the elements of B:  A, B: (A  B  A)  (A  B  B) 예제 : {a,b,c}  {2,3} = {a,b,c,2,3} {2,3,5}  {3,5,7} = {2,3,5,3,5,7} ={2,3,5,7} Set Operations

4. Discrete Mathematics by Yang-Sae Moon Page 3 Intersection Operator ( 교집합 ) For sets A, B, their intersection A  B is the set containing all elements that are simultaneously in A and (“  ”) in B. (A 와 B 양쪽 모두에 속하는 원소들의 집합 ) Formally, A  B = {x | x  A  x  B}. Note that A  B is a subset of A and it is a subset of B:  A, B: (A  B  A)  (A  B  B) 예제 : {a,b,c}  {2,3} =  {2,4,6}  {3,4,5} = {4} Set Operations

5. Discrete Mathematics by Yang-Sae Moon Page 4 Disjointedness ( 서로 소 ) Two sets A, B are called disjoint (i.e., unjoined) iff their intersection is empty. (A  B=  ) ( 교집합이 공집합이면 이들 두 집합은 서로 소라 한다.) Example: the set of even integers is disjoint with the set of odd integers. Set Operations

6. Discrete Mathematics by Yang-Sae Moon Page 5 Inclusion-Exclusion Principle ( 포함 - 배제 원리 ) How many elements are in A  B? |A  B| = |A|  |B|  |A  B| Example: How many students are on our class email list? Consider set E  I  M, I = {s | s turned in an information sheet} M = {s | s sent the TAs their email address} Some students did both! |E| = |I  M| = |I|  |M|  |I  M| 중복 제거 Set Operations

7. Discrete Mathematics by Yang-Sae Moon Page 6 Set Difference ( 차집합 ) For sets A, B, the difference of A and B, written A  B, is the set of all elements that are in A but not B. (A 에는 속하나 B 에는 속하지 않는 원소들의 집합 ) Formally, A  B =  x  x  A  x  B  예제 : {1, 2, 3, 4, 5, 6}  {2, 3, 5, 7, 11} = {1, 4, 6} Z  N  {…, -1, 0, 1, 2, … }  {0, 1, … } = {x | x is an integer but not a natural number} = {x | x is a negative integer} = {…, -3, -2, -1} Set Operations

8. Discrete Mathematics by Yang-Sae Moon Page 7 Set Difference – Venn Diagram A  B is what’s left after B “takes a bite out of A” Set A Set B Set A  B Chomp! ( 갉아먹어 !) Set Operations

9. Discrete Mathematics by Yang-Sae Moon Page 8 Set Complements ( 여집합 ) The domain can itself be considered a set, call it U. ( 정의역 자체도 집합이다.) We say that for any set A  U, the complement of A, written A, is the complement of A w.r.t. U, i.e., it is U  A. 예제 : If U=N, {3, 5} = {0, 1, 2, 4, 6, 7, …} An equivalent definition, when U is clear: A =  x  x  A  A U A Set Operations

10. Discrete Mathematics by Yang-Sae Moon Page 9 Set Identities ( 집합의 항등 ) (1/2) Identity Name A   = A A  U = A Identity laws A  U = U A   =  Domination laws A  A = A A  A = A Idempotent laws A = ADouble complement law A  B = B  AA  B = B  AA  B = B  AA  B = B  A Commutative laws A  (B  C) = (A  B)  C A  (B  C) = (A  B)  C Associative laws Set Operations

11. Discrete Mathematics by Yang-Sae Moon Page 10 Set Identities ( 집합의 항등 ) (2/2) Identity Name A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) Distributive laws A  B = A  BA  B = A  BA  B = A  BA  B = A  B De Morgan’s laws A  (A  B) = A A  (A  B) = A Absorption laws A  A = U A  A =  Complement laws Set Operations

12. Discrete Mathematics by Yang-Sae Moon Page 11 Proving Identity Sets ( 항등의 증명 ) To prove statements about sets, of the form A = B, here are three useful techniques: ( 집합의 항등 증명 방법에는 …) Method 1: Prove A  B and B  A separately. Method 2: Use set builder notation & logical equivalences. ( 조건 제시법과 논리적 동치 관계 활용 ) Method 3: Use a membership table. ( 구성원 표를 사용 ) Set Operations

13. Discrete Mathematics by Yang-Sae Moon Page 12 Proving Identity Sets (Example of Method 1) Example: Show A  (B  C)=(A  B)  (A  C). Show A  (B  C)  (A  B)  (A  C). Assume x  A  (B  C), & show x  (A  B)  (A  C). We know that x  A, and either x  B or x  C. (by assumption) −Case 1: x  B. Then x  A  B, so x  (A  B)  (A  C). −Case 2: x  C. Then x  A  C, so x  (A  B)  (A  C). Therefore, x  (A  B)  (A  C). Therefore, A  (B  C)  (A  B)  (A  C). Show (A  B)  (A  C)  A  (B  C). … Set Operations

14. Discrete Mathematics by Yang-Sae Moon Page 13 Proving Identity Sets (Example of Method 2) Example: Show (A  B)=A  B. (A  B)= {x | x  A  B} = {x | ¬(x  A  B)} = {x | ¬(x  A  x  B)} = {x | x  A  x  B} = {x | x  A  x  B} = {x | x  A  B} = A  B Set Operations

15. Discrete Mathematics by Yang-Sae Moon Page 14 Proving Identity Sets (Example of Method 3) Membership tables ( 구성원 표 ) Just like truth tables for propositional logic. ( 명제의 진리표와 유사 ) Columns for different set expressions. ( 열은 집합 표현을 나타냄 ) Rows for all combinations of memberships in constituent sets. Use “1” to indicate membership in the derived set, “0” for non-membership. ( 행에는 집합의 원소이면 1, 아니면 0 을 표시 ) Prove equivalence with identical columns. ( 두 컬럼이 동일함을 보임 ) Example: Prove (A  B)  B = A  B. Set Operations

16. Discrete Mathematics by Yang-Sae Moon Page 15 Generalized Unions and Intersections Since union & intersection are commutative and associative, we can extend them from operating on ordered pairs of sets (A, B) to operating on sequences of sets (A 1, …, A n ) ( 합집합 및 교집합은 교환 및 결합법칙이 성립하므로, 두 집합에 대한 연산을 확 장하여 세 개 이상의 집합에 대해서도 연산 적용이 가능하다.) Examples of generalized unions & intersections A = {0, 2, 4, 6, 8}, B = {0, 1, 2, 3, 4}, C = {0, 3, 6, 9} A  B  C = {0, 1, 2, 3, 4, 6, 8, 9} A  B  C = {0} Set Operations

17. Discrete Mathematics by Yang-Sae Moon Page 16 Generalized Union ( 합집합의 일반화 ) Binary union operator: A  B n-ary union: A 1  A 2  …  A n = ((…((A 1  A 2 )  …)  A n ) (grouping & order is irrelevant) “Big U” notation: Or for infinite sets of sets: Set Operations

18. Discrete Mathematics by Yang-Sae Moon Page 17 Generalized Intersection ( 교집합의 일반화 ) Binary intersection operator: A  B n-ary intersection: A 1  A 2  …  A n  ((…((A 1  A 2 )  …)  A n ) (grouping & order is irrelevant) “Big Arch” notation: Or for infinite sets of sets: Set Operations

19. Discrete Mathematics by Yang-Sae Moon Page 18 컴퓨터에서 집합의 표현 표현 방법 전체 집합 U 가 유한집합이라 가정하고, U 의 원소들을 a 1, a 2, …, a n 과 같이 순 서를 매긴다. U 의 부분집합 A 를 길이 n 의 비트열 (bit string) 로 표현한다. −i th bit = 1 if a i  A, 0 otherwise 예제 : U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 홀수 정수를 포함하는 부분집합을 표현하는 비트열 = 1010101010 짝수 정수를 포함하는 부분집합을 나타내는 비트열 = 0101010101 집합 연산과 컴퓨터 연산의 관계 합집합 (union) = Bitwise OR 교집합 (intersection) = Bitwise AND 여집합 (complement) = Bitwise NOT Set Operations