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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

이산수학 (Discrete Mathematics)  집합 연산 (Set Operations). 2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세. 2. 5. 3. 7. Union Operator ( 합집합 ). Set Operations.

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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

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

  2. 2 5 3 7 Union Operator (합집합) Set Operations • For sets A, B, theirunionAB is the set containing all elements that are either in A, or (“”) in B (or, of course, in both). (A또는 B에 속하거나 양쪽에 모두 속하는 원소들의 집합) • Formally, AB = {x | xAxB}. • Note that AB contains all the elements of Aandit 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}

  3. 3 2 4 6 5 Intersection Operator (교집합) Set Operations • For sets A, B, their intersectionAB is the set containing all elements that are simultaneously in A and (“”) in B. (A와 B양쪽 모두에 속하는 원소들의 집합) • Formally, AB = {x | xAxB}. • Note that AB is a subset of Aand 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}

  4. Disjointedness (서로 소) Set Operations 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.

  5. Inclusion-Exclusion Principle (포함-배제 원리) Set Operations • 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

  6. Set Difference (차집합) Set Operations • For sets A, B, the differenceof A and B, written AB, is the set of all elements that are in A but not B. (A에는 속하나 B에는 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}

  7. Chomp! (갉아먹어!) SetAB Set Difference – Venn Diagram Set Operations AB is what’s left after B “takes a bite out of A” Set A Set B

  8. Set Complements (여집합) Set Operations 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 A U

  9. Set Identities (집합의 항등) (1/2) Set Operations

  10. Set Identities (집합의 항등) (2/2) Set Operations

  11. Proving Identity Sets (항등의 증명) Set Operations • 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)

  12. Proving Identity Sets (Example of Method 1) Set Operations 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). …

  13. Proving Identity Sets (Example of Method 2) Set Operations 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

  14. Proving Identity Sets (Example of Method 3) Set Operations • 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 0을 표시) • Prove equivalence with identical columns. (두 컬럼이 동일함을 보임) • Example: Prove (AB)B = AB.

  15. Generalized Unions and Intersections Set Operations • 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 (A1, …, An) (합집합 및 교집합은 교환 및 결합법칙이 성립하므로, 두 집합에 대한 연산을 확장하여 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}

  16. Generalized Union (합집합의 일반화) Set Operations • Binary union operator: AB • n-ary union: A1A2…An = ((…((A1 A2) …) An) (grouping & order is irrelevant) • “Big U” notation: • Or for infinite sets of sets:

  17. Generalized Intersection (교집합의 일반화) Set Operations • Binary intersection operator: AB • n-ary intersection: A1A2…An((…((A1A2)…)An) (grouping & order is irrelevant) • “Big Arch” notation: • Or for infinite sets of sets:

  18. 컴퓨터에서 집합의 표현 Set Operations • 표현 방법 • 전체 집합 U가 유한집합이라 가정하고, U의 원소들을 a1, a2, …, an과 같이 순서를 매긴다. • U의 부분집합 A를 길이 n의 비트열(bit string)로 표현한다. • ith bit = 1 if ai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

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