Section 6.1 Sets and Set Operationsmayaj/Chapter6_Sec6.1_6... · 2. A new state employee is...

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

write a /2 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 � 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 2 B

(b) {x, y, z} ⇢ B

(c) {u, w} /2 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 Fall 2017, 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 A

c. Below are venn diagrams

illustrating the sets Ac and B

c.

A

cB

c

4. If A and B are two subsets of a universal set U , illustrate the sets Ac \B and A\B

c 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 [ A

c = U e. A \ A

c = ?

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 = A

c \B

c

(A \ B)c = A

c [B

c

5. Write venn diagrams to represent each of the following sets.

(a) A [ B

c

(b) A

c \ B

c


6. Write venn diagrams to represent each of the following sets.

(a) A \ B \ C

c

(b) A

c [ 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


-91-6,42/5,711,13174

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 \ B

c)

(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 \C

c) = 9. Find the following

values.

(a) n((A [ B [ C)c)

(b) n(Ac \ B

c \ 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

de

f

Basketball

g

h

a =

b =

c =

d =

e =

f =

g =

h =


32

AIMEE;I!IhtIn÷Iu . *

= 32

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 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?


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 di↵erent license plates are possible?

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


@

175,760€

H56,Oo9OJx

624,00£

(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?


15,120¥

5,@

435,45€

648¥

:@

Section 6.1 Sets and Set Operationsmayaj/Chapter6_Sec6.1_6... · 2. A new state employee is...

Documents

Transcript of Section 6.1 Sets and Set Operationsmayaj/Chapter6_Sec6.1_6... · 2. A new state employee is...