2. 2015 년 봄학기 강원대학교 컴퓨터과학전공 문양세 이산수학 (Discrete Mathematics) 집합과 연산 (Sets and Operations)

3. Discrete Mathematics by Yang-Sae Moon Page 2 강의 내용 집합 (Sets) 집합 연산 (Set Operations) Sets and Operations

4. Discrete Mathematics by Yang-Sae Moon Page 3 Introduction to Sets 정의 집합 (set) 이란 순서를 고려하지 않고 중복을 고려하지 않는 객체 (object) 들의 모임이다. A set is a new type of structure, representing an unordered collection (group) of zero or more distinct (different) objects. Basic notations for sets For sets, we’ll use variables S, T, U, … We can denote a set S in writing by listing all of its elements in curly braces { and }: {a, b, c} is the set of whatever three objects are denoted by a, b, c. Set builder notation: {x|P(x)} means the set of all x such that P(x). Sets 원소 나열법 ( 예 : {…, -3, -2, -1, 0, 1, 2, 3, …}) 조건 제시법 ( 예 : {x | x is an integer})

5. Discrete Mathematics by Yang-Sae Moon Page 4 Basic Properties of Sets Sets are inherently unordered: ( 순서가 중요치 않다 !) No matter what objects a, b, and c denote, {a, b, c} = {a, c, b} = {b, a, c} = {b, c, a} = {c, a, b} = {c, b, a}. All elements are distinct (unequal); multiple listings make no difference! ( 중복은 의미가 없다 !) If a=b, then {a, b, c} = {a, c} = {b, c} = {a, a, b, a, b, c, c, c, c}. This set contains at most two elements! Sets

6. Discrete Mathematics by Yang-Sae Moon Page 5 Definition of Set Equality Two sets are declared to be equal if and only if they contain exactly the same elements. ( 동일한 원소들로 이루어진 두 집합은 동일하다.) In particular, it does not matter how the set is defined or denoted. ( 집합의 equality 에서 정의나 표현은 중요치 않다.) For example: The set {1, 2, 3, 4} = {x | x is an integer where x > 0 and x 0 and <25} Sets

7. Discrete Mathematics by Yang-Sae Moon Page 6 Infinite Sets ( 무한 집합 ) Conceptually, sets may be infinite (i.e., not finite, without end, unending). ( 집합은 무한할 수 있다.  무한집합 ) Symbols for some special infinite sets: N = {0, 1, 2, …} The Natural numbers. ( 자연수의 집합 ) Z = {…, -2, -1, 0, 1, 2, …} The Zntegers. ( 정수의 집합 ) Z + = {1, 2, 3, …} The positive integers. ( 양의 정수의 집합 ) Q = {p/q | p  Z, q  Z, q  0} The rational numbers. ( 유리수의 집합 ) R = The Real numbers, such as 3.141592… ( 실수의 집합 ) Infinite sets come in different sizes! ( 무한집합이라도 크기가 다를 수 있다.)  We will cover it later. Sets

8. Discrete Mathematics by Yang-Sae Moon Page 7 Venn Diagrams Sets

9. Discrete Mathematics by Yang-Sae Moon Page 8 원소 (Elements or Members) x  S (“x is in S ”) is the proposition that object x is an  lement or member of set S. (x 는 S 의 원소이다.) e.g. 3  N, ‘a’  { x | x is a letter of the alphabet} Can define set equality in terms of  relation: ( 원소 기호를 사용한 두 집합의 동치 증명 ) S = T   x(x  S  x  T)   x(x  T  x  S)   x(x  S  x  T) “Two sets are equal iff they have all the same members.” x  S   (x  S)  “x is not in S” (x 는 S 의 원소가 아니다.) Sets

10. Discrete Mathematics by Yang-Sae Moon Page 9 공집합 (Empty Set)  (“null”, “the empty set”) is the unique set that contains no elements. ( 공집합 (  ) 이란 원소가 하나도 없는 유일한 집합이다.)  = {} = {x|False}   {  } 공집합 공집합을 원소로 하는 집합 Sets

11. Discrete Mathematics by Yang-Sae Moon Page 10 Subsets( 부분집합 ) and Supersets( 모집합 ) S  T (“S is a subset of T”) means that every element of S is also an element of T. (S 의 모든 원소는 T 의 원소이다.) S  T   x (x  S  x  T)   S, S  S ( 모든 집합은 공집합과 자기 자신을 부분집합으로 가진다.) S  T (“S is a superset of T”) means T  S. Note S = T  (S  T)  (S  T) S  T means  (S  T), i.e.  x(x  S  x  T) Sets

12. Discrete Mathematics by Yang-Sae Moon Page 11 Proper Subsets & Supersets S  T (“S is a proper subset of T”) means that S  T but T  S. Similar for S  T. (S 가 T 에 포함되나 T 는 S 에 포함되지 않으면, S 를 T 의 진부분집합이라 한다.) S T Venn Diagram equivalent of S  T Example: {1,2}  {1,2,3} Sets

13. Discrete Mathematics by Yang-Sae Moon Page 12 Sets are Objects, Too! The objects that are elements of a set may themselves be sets. ( 집합 자체도 객체가 될 수 있고, 따라서 집합도 다른 집합의 원소가 될 수 있다.) For example, let S={x | x  {1,2,3}} then S={ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} } Note that 1  {1}  {{1}} !!!! Sets

14. Discrete Mathematics by Yang-Sae Moon Page 13 Cardinality and Finiteness |S| (read “the cardinality of S”) is a measure of how many different elements S has. (|S| 는 집합 S 의 원소 중에서 서로 다른 값을 가지는 원소의 개수이다.) E.g., |  |=0, |{1,2,3}| = 3, |{a,b}| = 2, |{{1,2,3},{4,5}}| = ____ If |S|  N, then we say S is finite. Otherwise, we say S is infinite. What are some infinite sets we’ve seen? Sets

15. Discrete Mathematics by Yang-Sae Moon Page 14 Power Sets ( 멱집합 ) The power set P(S) of a set S is the set of all subsets of S. P(S) = {x | x  S} (P(S) 는 집합 S 의 모든 부분집합을 원소로 하는 집합 ) Examples: P({a,b}) = { , {a}, {b}, {a,b}}. P(  ) = {  }, P({  }) = { , {  }}. Sometimes P(S) is written 2 S. Note that for finite S, |P(S)| = |2 S |= 2 |S|. It turns out that |P(N)| > |N|. There are different sizes of infinite sets! ( 자연수 집합의 power set 의 크기가 자연수 집합보다 크다 ?!) Sets

16. Discrete Mathematics by Yang-Sae Moon Page 15 Ordered n-tuples Ordered n-tuples are like sets, except that duplicates matter, and the order makes a difference. (Ordered n-tuple 에서는 원소의 중복이 허용되고, 순서도 차이를 나타낸다.) For n  N, an ordered n-tuple or a sequence of length n is written (a 1, a 2, …, a n ). The first element is a 1, etc. Note (1, 2)  (2, 1)  (2, 1, 1). Empty sequence, singlets, pairs, triples, quadruples, …, n-tuples. 순서가 중요함 ( 의미를 가짐 ) 중복이 허용됨 ( 의미를 가짐 ) Sets

17. Discrete Mathematics by Yang-Sae Moon Page 16 Cartesian Products For sets A, B, their Cartesian product A  B = {(a, b) | a  A  b  B }. (a 가 A 의 원소이고 b 가 B 의 원소인 모든 순서쌍 (pair) (a, b) 의 집합이다.) E.g. {a,b}  {1,2} = {(a,1),(a,2),(b,1),(b,2)} Note that for finite A, B, |A  B|=|A||B|. Note that the Cartesian product is not commutative:  AB: A  B=B  A (Cartesian product 는 교환법칙이 성립하지 않는다.) E.g. {a,b}  {1,2}  {1,2}  {a,b} = {(1,a),(1,b),(2,a),(2,b)} A 1  A 2  …  A n = {(a 1, a 2, …, a n ) | a i  A i, i = 1, 2, 3, …, n} Sets

18. Discrete Mathematics by Yang-Sae Moon Page 17 강의 내용 집합 (Sets) 집합 연산 (Set Operations) Sets and Operations

19. Discrete Mathematics by Yang-Sae Moon Page 18 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

20. Discrete Mathematics by Yang-Sae Moon Page 19 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

21. Discrete Mathematics by Yang-Sae Moon Page 20 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

22. Discrete Mathematics by Yang-Sae Moon Page 21 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

23. Discrete Mathematics by Yang-Sae Moon Page 22 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

24. Discrete Mathematics by Yang-Sae Moon Page 23 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

25. Discrete Mathematics by Yang-Sae Moon Page 24 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

26. Discrete Mathematics by Yang-Sae Moon Page 25 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

27. Discrete Mathematics by Yang-Sae Moon Page 26 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

28. Discrete Mathematics by Yang-Sae Moon Page 27 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

29. Discrete Mathematics by Yang-Sae Moon Page 28 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

30. Discrete Mathematics by Yang-Sae Moon Page 29 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

31. Discrete Mathematics by Yang-Sae Moon Page 30 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

32. Discrete Mathematics by Yang-Sae Moon Page 31 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

33. Discrete Mathematics by Yang-Sae Moon Page 32 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

34. Discrete Mathematics by Yang-Sae Moon Page 33 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

35. Discrete Mathematics by Yang-Sae Moon Page 34 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

36. Discrete Mathematics by Yang-Sae Moon Page 35 강의 내용 집합 (Sets) 집합 연산 (Set Operations) Sets and Operations