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Sets and Venn Diagrams

- A set is a collection of objects.
- There are three ways to describe a set
- We can use words.
- We can make a list.
- We can use set-builder notation.

Examples of Sets

- 1. Words
- N is the set of natural numbers or counting

numbers. - 2. List
- N 1, 2, 3,
- 3. Set-builder notation
- N x x ? N

Kinds of Sets

- A finite set has a limited number of members.
- Example The set of students in our Math class.
- An infinite set has an unlimited number of

members. - Example The set of integers.
- A well-defined set has a universe of objects

which are allowed into consideration and any

object in the universe is either an element of

the set or it is not.

Venn Diagrams

- One way to represent or visualize sets is to use

Venn diagrams

Universe or Universal Set

- Let U be the set of all students enrolled in

classes this semester.

U

- Let M be the set of all students enrolled in

Math this semester. - Let E be the set of all students enrolled in

English this semester.

U

E

M

Complement of a set

- Let C be the set of all students enrolled in

classes this semester, but who are not enrolled

in Math or English

U

E

M

C

Intersection (?)

- E ? M the set of students in Math AND English

U

M

E

Union (?)

- E ? M the set of students in Math OR English

U

M

E

Disjoint Sets

- Two sets with no elements in common are called

disjoint sets.

U

Female Students

Male Students

Subset (?)

- X is a subset of Y if and only if every member of

X is also a member of Y.

U

Students in a Math class

Algebra Students

Example

- A survey of 100 students revealed that 82 were

in Math and 65 were in English. How many students

are taking both Math and English? All 100

students are either in Math or English.

U

Solution

- 82 65 147 82 47 35
- 147 100 47 65 47 18

U

Math

English

47

35

18

Example

- The manager at a local Country -Western station

reviewed the songs played during one 3-hour

program on her station. - 12 songs were about a truck driver who is in love

while in prison. - 13 songs talked about a prisoner in love.
- 28 songs talked about a person in love.
- 18 songs were about a truck driver in love.
- 3 songs were about truck drivers in prison who

are not in love. - 8 songs talked about people who are not in

prison, are not in love and don't drive a truck. - 16 songs were about truck drivers who are not in

prison. - 2 songs were about people in prison who are not

in love and are not truck drivers.

- Draw a Venn diagram.
- Use your Venn Diagram to answer the following

questions. - a) How many songs were about truck drivers?
- b) How many songs were about prisoners?
- c) How many songs were about truck drivers in

prison? - d) How many songs are about people in love

who are not truck drivers and not in prison? - e) How many songs did the station manager

review?

- Venn Diagram

Truck Driver

In Love

9

6

10

12

1

3

2

In Prison

8

Answers

- a. How many songs were about truck drivers? 31
- b. How many songs were about prisoners? 18
- c. How many songs were about truck drivers in

prison? 15 - d. How many songs are about people in love who

are not truck drivers and not in prison? 9 - e. How many songs did the station manager review?

51

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