Section 6.1 Sets and Set Operations

Home , Union (set theory), Universal set, Venn diagram

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 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 { }⇢

(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 [ [

-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 { | }

(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. How many three-digit numbers can be formed from the numerals in the set 1, 2, 3, 4 if the { } following is true?

(a) Repetition of digits is allowed. @

(b) Repetition of digits is not allowed.

6. Astatemakeslicenseplateswiththreelettersfollowedbyfourdigits.

(a) How many license plates are possible? 175,760€

(b) If no repetition of the letters is permitted, how many di↵erent license plates are possible?

(c) If no repetition of letters or digits is permitted, how many di↵erent license plates are possible? H56,Oo9OJx 624,00£

13 Fall 2017, Maya Johnson (d) How many license plates have no repetition of letters or digits and begin with a vowel?

15,120¥ n-Factorial (n!) For any natural number n, n!=n(n 1)(n 2) 3 2 1 ··· · · 0! = 1 7. Find 3!, 4! and 7! 5,@

8. Five men and ten women are to line up for a picture with the ﬁve men in the middle. How many ways can this be done? (Assume there are ﬁve women on each side of the group of men.)

435,45€

9. An exam consits of three true/false questions followed by four multiple choice questions each with 3answers.

(a) How many ways can a student answer the exam if they answer all of the questions?

648¥ (b) How many ways can a student answer the exam if they can leave true/false questions blank?

14 Fall 2017, Maya Johnson