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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

이산수학 (Discrete Mathematics)  집합 (Set). 2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세. Introduction to Sets. Sets. 정의 집합 ( set ) 이란 순서를 고려하지 않고 중복을 고려하지 않는 객체 (object) 들의 모임이다 .

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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

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

  2. Introduction to Sets 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. • Setbuilder notation: {x|P(x)} means the set of all x such that P(x). 원소 나열법 (예: {…, -3, -2, -1, 0, 1, 2, 3, …}) 조건 제시법 (예: {x | x is an integer})

  3. Basic Properties of Sets 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!

  4. Definition of Set Equality Sets 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 equality에서 정의나 표현은 중요치 않다.) For example: The set {1, 2, 3, 4} = {x | x is an integer where x > 0 and x < 5 } = {x | x is a positive integer whose square is >0 and <25}

  5. Infinite Sets (무한 집합) Sets • 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.

  6. Venn Diagrams Sets 2 0 4 6 8 Even integers from 2 to 9 -1 1 3 5 7 9 Odd integers from 1 to 9 Primes <10 Positive integers less than 10 Integers from -1 to 9

  7. 원소(Elements or Members) Sets • xS (“x is in S”) is the proposition that object x is an lement or member of set S. (x는 S의 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” (x는 S의 원소가 아니다.)

  8. 공집합(Empty Set) Sets  (“null”, “the empty set”) is the unique set that contains no elements. (공집합()이란 원소가 하나도 없는 유일한 집합이다.)  = {} = {x|False}   {} 공집합을 원소로 하는 집합 공집합

  9. Subsets(부분집합) and Supersets(모집합) Sets S  T (“S is a subset of T”) means that every element of S is also an element of T. (S의 모든 원소는 T의 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)

  10. Proper Subsets & Supersets Sets S  T (“S is a proper subset of T”) means that S  T but T  S. Similar for S  T. (S가 T에 포함되나 T는 S에 포함되지 않으면, S를 T의 진부분집합이라 한다.) Example:{1,2}  {1,2,3} S T Venn Diagram equivalent of S  T

  11. Sets are Objects, Too! Sets 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}} !!!!

  12. Cardinality and Finiteness Sets |S| (read “the cardinality of S”) is a measure of how many different elements S has. (|S|는 집합 S의 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? 2 N Z R

  13. Power Sets (멱집합) Sets • 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 S의 모든 부분집합을 원소로 하는 집합) • Examples: • P({a,b}) = {, {a}, {b}, {a,b}}. • P() = {}, P({}) = {, {}}. • Sometimes P(S) is written 2S. Note that for finite S, |P(S)| = |2S|= 2|S|. • It turns out that |P(N)| > |N|. There are different sizes of infinite sets! (자연수 집합의 power set의 크기가 자연수 집합보다 크다?!)

  14. Ordered n-tuples Sets Ordered n-tuples are like sets, except that duplicates matter, and the order makes a difference. (Ordered n-tuple에서는 원소의 중복이 허용되고, 순서도 차이를 나타낸다.) For nN, an ordered n-tuple or a sequenceoflength n is written (a1, a2, …, an). The first element is a1, etc. Note (1, 2)  (2, 1)  (2, 1, 1). Empty sequence, singlets, pairs, triples, quadruples, …, n-tuples. 중복이 허용됨(의미를 가짐) 순서가 중요함(의미를 가짐)

  15. Cartesian Products Sets • For sets A, B, their Cartesian product A  B = {(a, b) | aA  bB }. (a가 A의 원소이고 b가 B의 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 product는 교환법칙이 성립하지 않는다.) • E.g. {a,b}{1,2}  {1,2}{a,b} = {(1,a),(1,b),(2,a),(2,b)} • A1  A2  …  An = {(a1, a2, …, an) | aiAi, i = 1, 2, 3, …, n}

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