chapter 6 section 6.1 sets and set operations

Chapter 6

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, if a is not an element of a set A, then we

write a /∈ A.

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, written A = B, if and only if they have exactly the

same elements.

2. Let A = {a, e, l, t, r}. Which of the following sets are equal to A? (Choose all that apply.)

(a) {x | x is a letter of the word latter}

(b) {x | x is a letter of the word later}

(c) {x | x is a letter of the word late}

(d) {x | x is a letter of the word rated}

(e) {x | x is a letter of the word relate}

Subset If every element of a set A is also an element of a set B, then we say that A is a subset

of B and we write A ⊆ B.

Note: If we write A ⊂ B, then this means that A is a proper subset of B, without the possibility

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}, determine whether the following statements are true or false.

(a) x, y ∈ B

(b) {x, y, z} ⊂ B

(c) {u,w} /∈ A

(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 ∅. The set of all elements under discussion is called the

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, written A∪B, is the set of all elements

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 Summer 2018, ©Maya Johnson

Set Intersection Let A and B be sets. The intersection of A and B, written A∩B, is the set

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 , then the set of all elements

in U that are not in A is called the complement of A and is denoted Ac. Below are venn diagrams

illustrating the sets Ac and Bc.

Ac Bc

4. If A and B are two subsets of a universal set U , illustrate the sets Ac ∩B and A∩Bc using venn

diagrams.


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, and C 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


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 = ∅.


7. Let U denote the set of all senators in Congress and let

D = {x is in U | x is a Democrat}

R = {x is in U | x is a Republican}

F = {x is in U | x is a female}

L = {x is in U | x is a lawyer}.

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?


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}, and C

= {-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


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) is the number of elements in the set A. If A 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 , then n(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 | x is an even number from 1 to 10}

(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, and n(A ∩B) = 6, find n(A).


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, where m is any nonnegative integer.

Then the number of subsets of A is 2m. The number of proper 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?


5. Let A, B, and C be sets in a universal set U . We are given n(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)


6. Use the following information to determine the number of people in each region of the Venn

Diagram.

A group of 295 students were asked which of these sports they participated in during high school.

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.

Tennis

a b

Track

c

d

ef

Basketball

g

h

a =

b =

c =

d =

e =

f =

g =

h =


Section 6.3 The Multiplication Principle

The Multiplication Principle

Suppose there are m ways to do a task T1 and there are n ways to do a task T2. Then there are m · nways 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

N1N2 · · ·Nm.

2. A new state employee is offered a choice of eight basic health plans, five dental plans, and two

vision care plans. How many different 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 different

license plates can be issued using this configuration?


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. A state makes license plates with three letters followed by four digits.

(a) How many license plates are possible?

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

(c) If no repetition of letters or digits is permitted, how many different license plates are possible?


(d) How many license plates have no repetition of letters or digits and begin with a vowel?

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!

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.)

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

3 answers.

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

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

(c) How many ways can a student answer the exam if they can leave any of the questions blanks?


Section 6.4 Permutations and Combinations: Part 1

Permutations

1. How many ways can you arrange three people in a line?

2. Five people are waiting to take a picture. How many ways can you arrange three of them in a

line to take the picture?

Reminder of Factorials

n-Factorial (n!) For any natural number n,

n! = n(n− 1)(n− 2) · · · 3 · 2 · 1

0! = 1

Calculator Steps: Enter the number followed by MATH , scroll right to PRB and click 4 then

click ENTER .

Permutations of n Distinct Objects:

The number of permutations of n distinct objects taken r at a time is

P (n, r) =n!

(n− r)!

Calculator Steps: Enter the first number followed by MATH , scroll right to PRB and click 2

then enter the second number and click ENTER .

Note: If n = r, then P (n, n) = n!

3. Compute P (3, 3) and P (5, 3). How do these two permutations relate to the answers in examples

1 and 2?


4. In how many ways can the names of nine candidates for political office be listed on a ballot?

Note: In this class, I tend to use the Multiplication Principle interchangeably with Permutations.

5. Rework the previous example using the Multiplication Principle instead of Permutations.

6. A company car that has a seating capacity of eight is to be used by eight employees who have

formed a car pool. If only three of these employees can drive, how many possible seating arrange-

ments are there for the group?

7. There are four families attending a concert together. Each family consists of 1 male and 2 females.

In how many ways can they be seated in a row of twelve seats if

(a) There are no restrictions?

(b) Each family is seated together?


(c) The members of each gender are seated together?

8. At a college library exhibition of faculty publications, two mathematics books, four social science

books, and three biology books will be displayed on a shelf. (Assume that none of the books are

alike.)

(a) In how many ways can the nine books be arranged on the shelf?

(b) In how many ways can the nine books be arranged on the shelf if books on the same subject

matter are placed together?

Permutations of n Objects, Not all Distinct:

Given a set of n objects in which n1 objects are alike and of one kind, n2 objects alike and of

another kind, . . . , and nm objects are alike and of yet another kind, so that

n1 + n2 + · · ·+ nm = n

then the number of permutations of these n objects taken n at a time is given by

n!

n1!n2! · · ·nm!


9. Find the number of distinguishable arrangements of each of the following “words.”

(a) acdbens

(b) baaaben

(c) aaabbba

10. A toy chest contains four identical blue blocks, three identical yellow blocks, and seven identical

red blocks. How many distinguishable arrangements of these blocks can be made?

11. Suppose that in the previous example, the blocks of the same color are numbered, so that the

blue blocks are numbered 1 through 4, the yellow blocks are numbered 1 through 3 and the red

blocks are numbered 1 through 7. Note that this means that blocks of the same color are no

longer identical.

(a) How many distinguishable arrangements of these blocks can be made?

(b) How many distinguishable arrangements of these blocks can be made if blocks of the same

color should stay together?


Section 6.4 Permutations and Combinations Part 2

Combinations

Question:

Suppose we want to choose three people from a group of four people and we do not care about the order

in which we do this, that is, we will not be arranging the people we choose in any particular order.

How do we do this?

Answer:

Suppose we number the people from 1 through 4 and think of the set A = {1, 2, 3, 4}. To answer this

question we will count how many subsets of size 3 there are of this set...

Combinations of n Distinct Objects:

The number of combinations of n distinct objects taken r at a time is given by

C(n, r) =n!

r!(n− r)!(where r ≤ n)

Calculator Steps: Enter the first number followed by MATH , scroll right to PRB and click 3 then

enter the second number and click ENTER .

1. Compute C(4, 3) and C(10, 5).

Language If a problem uses the word “and” (∩) then you need to multiply the results. If a

problem uses the word “or” (∪) then you need to add the results.


2. In how many ways can a subcommittee of six be chosen from a Senate committee of six Democrats

and five Republicans if

(a) All members are eligible?

(b) The subcommittee must consist of three Republicans and three Democrats?

3. In how many different ways can a panel of 12 jurors and 2 alternates be chosen from a group of

16 prospective jurors?

4. A student planning her curriculum for the upcoming year must select one of four business courses,

one of four mathematics courses, two of seven elective courses, and either one of five history

courses or one of three social science courses. How many different curricula are available for her

consideration?


5. From a shipment of 25 transistors, 6 of which are defective, a sample of 9 transistors is selected

at random.

(a) In how many different ways can the sample be selected?

(b) How many samples contain exactly 3 defective transistors?

(c) How many samples contain no defective transistors?

(d) How many samples contain at least 5 defective transistors?

Complement Rule: Sometimes it is easier to ask how many ways there are of doing the

opposite (or complement) of what you want than it is to ask how many ways there are to do what

you want. So the complement rule is

# of Ways You Want = Total Ways - # of Ways You Don’t Want


6. A box contains 8 red marbles, 8 green marbles, and 10 black marbles. A sample of 12 marbles is

to be picked from the box.

(a) How many samples contain at least 1 red marble?

(b) How many samples contain exactly 4 red marbles and exactly 3 black marbles?

(c) How many samples contain exactly 7 red marbles or exactly 6 green marbles?


(d) How many samples contain exactly 5 green marbles or exactly 3 black marbles?

When To Use Permutations/Multiplication Principle or Combinations?

We use permutations/multiplication principle whenever order matters and we use combinations

whenever order does not matter.

� Keywords that suggest we use permutations/multiplication principle: distinguishable ar-

rangments, ordered list of names, distinct arrangments, arrange in a line, seated in a line or

seated in any arrangment

� Keywords that suggest we use combinations: choose a smaller group from a larger group,

select a committee or subcommittee, select a number of items, how many samples contain a

number of items or people

7. In how many ways can the names of three Republican and four Democratic candidates for political

office be listed on a ballot?

(a) Should we use combinations or should we use permutations/multiplication principle?

(b) How many ways can this be done?


8. A bag contains 5 red marbles and 5 blue marbles. How many ways can you select 6 marbles from

the bag?

(a) Should we use combinations or should we use permutations/multiplication principle?

(b) How many ways can this be done?

Note: Sometimes a problem requires using both permutations/multiplication principle and com-

binations.

9. Suppose we have 25 people on a committee. How many subcommittees contain one president,

one vice president and six cabinet members?

10. Twenty runners are competing in a half-marathon. How many ways can we award one 1st place

prize, one 2nd place prize, one 3rd place prize, and four 4th place prizes?


chapter 6 section 6.1 sets and set operations

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