Chapter 2 - Setsmath.utoledo.edu/~dgajews/1180_old/2-Sets.pdfSection 2.1, Slide 9 Representing Sets • A set is well-defined if we are able to tell whether any particular object is

Chapter 2 - Sets

Definitions

- A set is a collection of objects

- An element/member is an object in a set

Examples of sets

- The collection of whole numbers

- The collection of people who have been in a movie with Kevin Bacon

- The collection of people who wrote a whole book on a Thursday, speak multiple languages, and have a last name starting with ‘A’.

There are two common ways to list sets:

- Roster Notation lists every element in the set

- Set Builder provides a rule to find all elements in a set

Both notations are placed within braces { }

Roster Notation (lists every element)

Examples: { 1, 2, 3, 4, 5 }

{ Cincinnati, Cleveland, Columbus }

Ellipsis “…” can be used to show that elements continue in the same manner.

Examples:

{ 1, 2, 3, 4, 5, … , 10 }

{ 0, 1, 2, 3, 4, 5, 6, … }

{ … , -3, -2, -1, 0, 1, 2, 3, … }

{ Washington, Adams, … , Bush, Obama }

© 2010 Pearson Education, Inc. All rights reserved. Section 2.1, Slide 7

Representing Sets

• Set-builder notation:

•

Examples of sets in Set Builder Notation

A = { x : x is a whole number and 1 <= x and x <= 10 }

B = { x : x was in a movie with Kevin Bacon }

C = { x : x is a complete graph }


Representing Sets

• A set is well-defined if we are able to tell whether any particular object is an element of the set.

• Example: Which sets are well-defined?(a) (b)

{ }: is a winner of an Academy AwardA x x=

{ }: is tallT x x=

Pop Quiz!!!!!

How many elements are in the set

A) { x : x is a day of the week }

B) { 4, 6, 8, … 14 }

C) { x : x is a student at UT and x is a space alien }

How do we represent the following set in Roster Form?

{ x : x is a student at UT and x is a space alien }

{ }

A set with no entries is known as the empty set. It can also be written as

∅


Representing Sets

• Do ∅ and {∅} mean the same thing?

– ∅ is the empty set – a set with no members– {∅} is a set with a member object, namely, the

empty set

The previous example shows that is possible to have a set of sets. (And sets of sets of sets, and … )

Examples:

{ 1, { 1 } }

{ { 1, 2, 3, 4, 5, … } , {-1, -2, -3, -4, -5, … } }

{ { { 3, 5 }, { 1 } } , { Kevin Bacon } }


The Element Symbol

• Example:

means "is an element of" means "is an element of"not

∈∉

{ }{ }2,3,4,53

6 2,3,4,5∈∉

K4 { x : x is a complete graph }


Representing Sets

• Example: Consider female consumers living in the U.S. The universal set is

{ }: is a female cosumer living in the U.S.U x x=


Cardinal Number

• Example: State the cardinal number of the set.

{ } { } { }{ }1,2,3 , 1,4,5 , 3X =

( ) 3

(the set contains 3 objects, each of which is also a set)

n XX

=

Examples:

{ x : x is an odd whole number } { 1, 3, 5, … }

{ x : x is an Euler path} { x : x is a Ham. path}

{ 1, 2, 3 } { 3, 2, 1 }

Section 2.2 Comparing two sets.

{ 1, 2, 3 } { x : x is a whole number }

{ 2, 4, 6, 8 } { … , -4, -2, 0, 2, 4, … }

{ 1, 2, 3, 4, 5, ... } { 1, 2, 3, … , 10}

∅ { 1, 2, 3 }


Venn Diagrams and Proper Subsets

• A Venn diagram is used to visualize relationships among sets.

• Here is the Venn diagram for A B.

The Venn Diagram of the different types of numbers

Real Numbers

Rational Numbers

Integers

Whole Numbers

Natural Numbers

What are all of the subsets of { a, b, c} ?

How many are there?


Venn Diagrams and Proper Subsets

• How many subsets exist for the given set?

{ }Bill, Gill, Jill, Will

42 2 16

A

k

=

= =


Equivalent Sets

• The sets {1, 2, 3} and {A, B, C} are equivalent because they both have 3 members.

• The sets { x : x is a pancreas in a UT student } and{ x : x is a medulla oblongata in a UT student } are

equivalent. Why?

Section 2.3 Operations on Sets

For numbers we have operations like: +, -, x, etc.

Sets also have operations on them.

First, we will look at Venn Diagrams for the 4 ways that two sets A and B can be related.

A and B could have no elements in common.

A B is a subset

U

A B

U

BA

A = B

A and B overlap, not equal but contain some of the same elements

U

A = B

U

A B

U

A B

The last Venn Diagram can be used to represent the others, so use it from now on.

U

A B

Different regions represent different types of elements.

I II III

IV

I = elements in A, not B III = elements in B, not A II = elements in A and B IV = elements not in A nor B

U

A B

Find the sets U, A, B.

1

2

3

45

6

U =A =B =

U

A B

Find the sets U, A, B.

1

2

3

45

6

U =A =B =

U

A B

Fill in the Venn diagram knowing U, A, B.

U = { 1, 2, 3, 4, 5, 6, 7 }A = { 1, 2, 3, 4 }B = { 1, 3, 5, 7 }


Union of Sets

• Example: Find the union of the two sets.

{ }{ }{ }

1, 3, 5, 6, 8

2, 3, 6, 7, 8

1, 2, 3, 5, 6, 7, 8

A

B

A B

=

=

=U


Intersection of Sets

• Example: Find the intersection of the two sets.

{ }{ }

{ }

1, 3, 5, 6, 8

2, 3, 6, 7, 8

3, 6, 8

A

B

A B

=

=

=∩


Set Complement

• Example: Given U, find the complement of A.

{ }{ }{ }

1, 2, 3, , 10

1, 3, 5, 7, 9

2, 4, 6, 8, 10

U

A

A

=

=

′ =


Set Difference

• Example: Find the difference.

{ } { }3, 6, 9, 12 : is an odd integer x x−


Order of Set Operations• Example: Let { } { }1,2,3, ,10 , : is even ,U E x x= =

{ } { }( ) ( )1,3,4,5,8 , and 1,2,4,7,8 .

Find .

B A

A B E A

= =

′ ′∩U U


Order of Set Operations

• Intersection distributes over union.

( ) ( ) ( ) A B C A B A C=∩ ∩ ∩U U


Order of Set Operations

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Chapter 2 - Setsmath.utoledo.edu/~dgajews/1180_old/2-Sets.pdfSection 2.1, Slide 9 Representing Sets • A set is well-defined if we are able to tell whether any particular object is