Sets and Operations Sets Set Operations Page 2

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

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). 원소 나열법 (예: {…, -3, -2, -1, 0, 1, 2, 3, …}) 조건 제시법 (예: {x | x is an integer}) Page 3 Discrete Mathematics by Yang-Sae Moon

Basic Properties of 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, c, c}. • This set contains at most two elements! Page 4 Discrete Mathematics by Yang-Sae Moon

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} Page 5 Discrete Mathematics by Yang-Sae Moon

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 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. Page 6 Discrete Mathematics by Yang-Sae Moon

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

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” Page 8 (x는 S의 원소가 아니다. ) Discrete Mathematics by Yang-Sae Moon

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 Page 9 Discrete Mathematics by Yang-Sae Moon

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 means (S T), i. e. x(x S x T) Page 10 Discrete Mathematics by Yang-Sae Moon

Proper Subsets & Supersets 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. ) S Example: {1, 2} {1, 2, 3} T Venn Diagram equivalent of S T Page 11 Discrete Mathematics by Yang-Sae Moon

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}} !!!! Page 12 Discrete Mathematics by Yang-Sae Moon

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? Page 13 Discrete Mathematics by Yang-Sae Moon

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 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 set의 크기가 자연수 집합보다 크다? !) Page 14 Discrete Mathematics by Yang-Sae Moon

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 sequence of length n is written (a 1, a 2, …, an). The first element is a 1, etc. Note (1, 2) (2, 1, 1). 중복이 허용됨(의미를 가짐) 순서가 중요함(의미를 가짐) Empty sequence, singlets, pairs, triples, quadruples, …, n -tuples. Page 15 Discrete Mathematics by Yang-Sae Moon

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} {a, b} = {(1, a), (1, b), (2, a), (2, b)} A 1 A 2 … An = {(a 1, a 2, …, an) | ai Ai, i = 1, 2, 3, …, n} Page 16 Discrete Mathematics by Yang-Sae Moon

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

Union Operator (합집합) Set Operations 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 B에 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, 7} ={2, 3, 5, 7} Page 18 Discrete Mathematics by Yang-Sae Moon

Intersection Operator (교집합) Set Operations For sets A, B, their intersection A B is the set containing all elements that are simultaneously in A and (“ ”) in B. (A와 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} Page 19 Discrete Mathematics by Yang-Sae Moon

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. Page 20 Discrete Mathematics by Yang-Sae Moon

Inclusion-Exclusion Principle (포함-배제 원리) Set Operations How many elements are in A B? |A B| = |A| |B| |A B| 중복 제거 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| Page 21 Discrete Mathematics by Yang-Sae Moon

Set Difference (차집합) Set Operations 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. (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} Page 22 Discrete Mathematics by Yang-Sae Moon

Set Difference – Venn Diagram Set Operations A B is what’s left after B “takes a bite out of A” Chomp! (갉아먹어!) Set A B Set A Set B Page 23 Discrete Mathematics by Yang-Sae Moon

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 U A Page 24 Discrete Mathematics by Yang-Sae Moon

Set Identities (집합의 항등) (1/2) Identity A A A =A U=U = A=A A=A A A B=B A (B C) = (A B) C Set Operations Name Identity laws Domination laws Idempotent laws Double complement law Commutative laws Associative laws Page 25 Discrete Mathematics by Yang-Sae Moon

Set Identities (집합의 항등) (2/2) Identity A (B C) = (A B) (A C) A A B=A B (A B) = A A A=U A A= Set Operations Name Distributive laws De Morgan’s laws Absorption laws Complement laws Page 26 Discrete Mathematics by Yang-Sae Moon

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) Page 27 Discrete Mathematics by Yang-Sae Moon

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, A (B C) (A B) (A C). Show (A B) (A C) A (B C). … Page 28 Discrete Mathematics by Yang-Sae Moon

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 B} =A B Page 29 Discrete Mathematics by Yang-Sae Moon

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. Page 30 Discrete Mathematics by Yang-Sae Moon

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 (A 1, …, An) (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} Page 31 Discrete Mathematics by Yang-Sae Moon

Generalized Union (합집합의 일반화) Set Operations Binary union operator: A B n-ary union: A 1 A 2 … An = ((…((A 1 A 2) …) An) (grouping & order is irrelevant) “Big U” notation: Or for infinite sets of sets: Page 32 Discrete Mathematics by Yang-Sae Moon

Generalized Intersection (교집합의 일반화) Set Operations Binary intersection operator: A B n-ary intersection: A 1 A 2 … An ((…((A 1 A 2) …) An) (grouping & order is irrelevant) “Big Arch” notation: Or for infinite sets of sets: Page 33 Discrete Mathematics by Yang-Sae Moon

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