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Section 6.1 Sets and Set Operations

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Section 6.1 Sets and Set Operations Section 6.1 Sets and Set Operations Sets A set is a well-defined collection of objects. The objects in this collection are called elements of the set. If a is an element of the set A then we write a A,ifa is not an element of a set A,thenwe 2 write a/A. 2 Roster and Set-Builder Notation Roster notation will be used most commonly in this class, and consists of listing the elements of a set in between curly braces. Set-builder notation is when a rule is used to define a definite property that an object must have in order to be in the set. 1. Let A be the set of all letters in the English alphabet. (a) Write A in roster notation and in set-builder notation. (b) Is the greek letter β an element of A? Set Equality Two sets A and B are equal,writtenA = B,ifandonlyiftheyhaveexactlythe same elements. 2. Let A = a, e, l, t, r . Which of the following sets are equal to A?(Chooseallthatapply.) { } (a) x xisaletterofthewordlatter { | } (b) x xisaletterofthewordlater { | } (c) x xisaletterofthewordlate { | } (d) x xisaletterofthewordrated { | } (e) x xisaletterofthewordrelate { | } Subset If every element of a set A is also an element of a set B,thenwesaythatA is a subset of B and we write A B. ✓ Note: If we write A B,thenthismeansthatA is a proper subset of B,withoutthepossibility ⇢ of equality. Therefore, for any set A, A is NOT a proper subset of itself. 3. If A = u, v, y, z and B = x, y, z ,determinewhetherthefollowingstatementsaretrueorfalse. { } { } (a) x, y B 2 (b) x, y, z B { }⇢ (c) u, w / A { } 2 (d) x, w A { }✓ The Empty and Universal Set The set that contains no elements is called the empty set and the symbol for the empty set is ?.Thesetofallelementsunderdiscussioniscalledthe universal set and is usually denoted by U. Note: The empty set is a subset of every set. That is, ? A where A is any set. ✓ Set Operations Set Union Let A and B be sets. The union of A and B,writtenA B,isthesetofallelements [ that belong to either A or B or both. This is like adding the two sets. Below is a Venn Diagram illustrating the set A B. [ A B [ A B 2 Fall 2017, Maya Johnson Set Intersection Let A and B be sets. The intersection of A and B,writtenA B,istheset \ of all elements that belong to both A and B. This is what the two sets have in common. Below is a venn diagram illustrating the set A B. \ A B \ A B Complement of a Set If U is a universal set and A is a subset of U,thenthesetofallelements in U that are not in A is called the complement of A and is denoted Ac.Belowarevenndiagrams illustrating the sets Ac and Bc. Ac Bc 4. If A and B are two subsets of a universal set U,illustratethesetsAc B and A Bc using venn \ \ diagrams. 3 Fall 2017, Maya Johnson Set Complementation If U is a universal set and A is a subset of U,then a. U c = ? b. ?c = U c. (Ac)c = A d. A Ac = U e. A Ac = ? [ \ Properties of Set Operations Let U be a universal set. If A, B,andC are arbitrary subsets of U,then A B = B A Commutative law for union [ [ A B = B A Commutative law for intersection \ \ A (B C)=(A B) C Associative law for union [ [ [ [ A (B C)=(A B) C Associative law for intersection \ \ \ \ A (B C)=(A B) (A C) Distributive law for union [ \ [ \ [ A (B C)=(A B) (A C) Distributive law for intersection \ [ \ [ \ De Morgan’s Laws Let A and B be sets. Then (A B)c = Ac Bc [ \ (A B)c = Ac Bc \ [ 5. Write venn diagrams to represent each of the following sets. (a) A Bc [ (b) Ac Bc \ 4 Fall 2017, Maya Johnson 6. Write venn diagrams to represent each of the following sets. (a) A B Cc \ \ (b) Ac B C [ [ Disjoint Sets Two sets A and B are disjoint if and only if they have no elements in common. That is, if A B = ?. \ 5 Fall 2017, Maya Johnson 7. Let U denote the set of all senators in Congress and let D= xisinU xisaDemocrat { | } R= xisinU xisaRepublican { | } F= xisinU xisafemale { | } L= xisinU xisalawyer . { | } Write the set that represents each statement. (a) The set of all Republicans who are female or are lawyers. (b) The set of all senators who are not Republicans or are lawyers Are the sets in parts (a) and (b) disjoint? 6 Fall 2017, Maya Johnson 8. Let U = -9, -6, -1, 2, 5, 7, 11, 13, 17, 19 , A = -9, -1, 5, 11, 17 , B = -6, 2, 7, 13, 19 ,andC { } { } { } = -9, -6, 2, 5, 13, 17 . Find each set using roster notation. { } (a) (A B) C \ [ (b) (A B C)c [ [ (c) (A B C)c \ \ -91-6,42/5,711,13174 7 Fall 2017, Maya Johnson Section 6.2 The Number of Elements in a Finite Set Counting Problems If a problem requires knowing the number of elements in a given set, then we call such a problem a Counting problem. Number of Elements in A If A is a set, then n(A)isthenumberofelementsinthesetA.IfA is a finite set, then we can simply count the number of elements in A to find n(A). Note: If U is a universal set and A is a subset of U,thenn(Ac)=n(U) n(A) − 1. Let the universal set U = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 . Find the following. { } (a) n(U) (b) n(Ac), where A = x xisanevennumberfrom1to10 { | } (c) n(B), where B = 1, 3, 9 { } (d) n(?) Addition Rule for Sets: Very Useful Formula If A and B are finite sets then n(A B)=n(A)+n(B) n(A B) [ − \ 2. If n(B)=13,n(A B)=24,andn(A B)=6,findn(A). [ \ 8 Fall 2017, Maya Johnson 3. In a survey of 272 people, a pet food manufacturer found that 69 owned a dog but not a cat, 28 owned a cat but not a dog, and 73 owned neither a dog or a cat. (a) How many owned both a cat and a dog? (b) How many owned a dog? Number of Subsets Suppose A is a set and that n(A)=m,wherem is any nonnegative integer. Then the number of subsets of A is 2m.Thenumberofproper subsets of A is 2m 1. − 4. Let A and B be subsets of a universal set U and suppose n(U)=48,n(A)=13,n(B)=23,and n(A B) = 8. Compute: \ (a) n(Ac B) \ (b) n(Bc) (c) n(Ac Bc) \ (d) How many subsets does A have? (e) How many proper subsets does A have? 9 Fall 2017, Maya Johnson 5. Let A, B,andC be sets in a universal set U.Wearegivenn(U)=66,n(A)=32,n(B)=33, n(C)=33,n(A B)=16,n(A C)=10,n(B C)=18,n(A B Cc) = 9. Find the following \ \ \ \ \ values. (a) n((A B C)c) [ [ (b) n(Ac Bc C) \ \ 10 Fall 2017, Maya Johnson 6. Use the following information to determine the number of people in each region of the Venn Diagram. Agroupof295studentswereaskedwhichofthesesportstheyparticipatedinduringhighschool. 44 students participated in all of these sports. 87 students participated in basketball and track. 39 students participated in basketball and tennis but not track. 79 students participated in track but not tennis. 155 students participated in basketball. 142 students did not participate in tennis. 103 students participated in exactly one sport. a = b = 32 Track Tennis c = a b c d = e d f e = g f = Basketball h g = h = ;I!IhtIn÷IuAIMEE . * = 32 11 Fall 2017, Maya Johnson Section 6.3 The Multiplication Principle The Multiplication Principle Suppose there are m ways to do a task T and there are n ways to do a task T .Thentherearem n 1 2 · ways of doing the task T1 followed by the task T2. 1. Four commuter trains and two express buses depart from city A to City B in the morning, and five commuter trains and five express buses operate on the return trip in the evening (from City B to City A). In how many ways can a commuter from City A to City B complete a daily round trip via bus and/or train? Generalized Multiplication Principle Suppose a task T1 can be done in N1 ways, a task T2 can be done in N2 ways,...,and, finally, a task Tm can be done in Nm ways. Then the number of ways of doing the tasks T1,T2,...,Tm in succession is given by the product N N N . 1 2 ··· m 2. A new state employee is o↵ered a choice of eight basic health plans, five dental plans, and two vision care plans. How many di↵erent health-care plans are there to choose from if one plan is selected from each category? 3. In recent years, a state has issued license plates using a combination of two letters of the alphabet followed by three digits, followed by another two letters of the alphabet. How many di↵erent license plates can be issued using this configuration? 12 Fall 2017, Maya Johnson 4. Complete the following. (a) How many seven-digit telephone numbers are possible if the first digit must be nonzero? (b) How many international direct-dialing numbers for calls within the United States and Canada are possible if each number consists of a 1 plus a three-digit area code (the first digit of which must be nonzero) and a number of the type described in part (a)? 5.
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