Section 2.1

Sets and Whole Numbers

Mathematics for Elementary School Teachers - 4th EditionO’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

How do you think the idea of numbers

developed?How could a child who doesn’t know how to

count verify that 2 sets have the same number of objects? That one set has more than another set?

Sets and Whole Numbers - Section 2.1A set is a collection of objectsor ideas that can be listed or

described

A = {a, e, i, o, u} C = {Blue, Red, Yellow}

A set is usually listed with a capital letterA set can be represented using braces { }

A set can also be represented using a circle

A = oi

eua C =

BlueRed

Yellow

Each object in the set is called an element of the set

C = BlueRed

YellowBlue is an element of set C

Blue C

Orange is not an element of set C

Orange C

Definition of a One-to-One CorrespondenceSets A and B have a one-to-one

correspondence if and only if each element of A is paired with exactly one element of B and each element of B is paired with exactly one element of A.

Set A

1

2

3

Set B

c

b

a

The order of the elements does not matter

Definition of Equal SetsSets A and B are equal sets if and only if each element of A is also an element of

B and each element of B is also an element of A

A = {Mary, Juan, Lan}B = {Lan, Juan, Mary}

Then, A = BSo equal sets are when both

sets contain the same elements - but the order of the elements

does not matter

Definition of Equivalent SetsSets A and B are equivalent sets if and

only if there is a one-to-one correspondence between A and B

Set A

onetwo

three

Set B

FrogCat

Dog

A~B

Definition of a Subset of a SetFor all sets A and B, A is a subset of B if and only if each element of A is also

an element of B

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 . . . }

W = {0, 1, 2, 3, 4, 5, . . . }

NaturalNumbers

Whole Numbers

Example: The Natural Numbers and the Whole Numbers

A B

N W

Set A

Set B

If set A contains elements that are not also in B, then set A is not a

subset of set B

Example:A = {dog, cat, fox, monkey, rabbit}

B = {dog, cat, fox, elephant, deer}

set A contains animals that are not in set B

A⊈B

Thus, A⊈B

Definition of a Proper Subset of a SetFor all sets A and B, A is a proper subset of B, if

and only if A is a subset of B and there is at least one element of B that is not an element of

A.

NaturalNumbers

Whole Numbers N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 . . . }

W = {0, 1, 2, 3, 4, 5, . . . }

Set A

Set B

N ⊂ W

A ⊂ B

The Universal Set, UThe Universal set is either given or

assumed from the context. If set A is the primary colors, then U could be assumed

to be the set of all colors

The universal set is generally shown in a venn diagram as a

rectangular area

RedBlue

Yellow

U

The Empty Set

A set with no elements

Symbols for the empty set: { } or ∅

Complement of a setThe complement of a set A is all the

elements in the universal set that are not in A

A

Finite Set

A set with a limited number of elements

Example: A = {Dog, Cat, Fish, Frog}

Infinite Set

A set with an unlimited number of elements

Example: N = {1, 2, 3, 4, 5, . . . }

Number of Elements in a Finite Set

To show the number of elements in a finite set we use the symbol: n(name of set)

Example: A = {Dog, Cat, Frog, Mouse}

n(A) = 4So, if two sets are equivalent (have the same number of elements) we use the

symbol:n(A) = n(B)

To show the empty set has no elements:n(∅) = 0 or n( { } ) = 0

Counting and Sets

“To determine the number of objects in a set we use the counting process to set up a one-to-one

correspondence between the number names and the objects in the set. That is, we say the number names in order and point at an object for each name. The last name said is the whole number of objects in the set.” (class text, p. 65)

“Counting is the process that enables people systematically to associate a whole number

with a set of objects.” (class text, p. 65)

A = {Dog, Cat, Frog, Mouse}B = { 1, 2, 3, 4 }

Less Than and Greater ThanFor whole numbers a and b and sets A and B, where n(A) = a and n(B) = b, a is less than b, (a<b), if and only if A is equivalent to a proper subset of B. Also, a is greater than b, (a>b),

whenever b<a.Example:

B = {dog, cat, fox, monkey, rabbit}

A = {dog, cat, fox, rabbit}n(A) = 4

n(B) = 5

Set A is equivalent to a proper subset of B

4 is less than 5 (4 < 5)

Important Subsets of the Whole Numbers

The Set of Whole Numbers

W = { 0, 1, 2, 3, 4, . . . }

N = { 1, 2, 3, 4, . . . }

E = { 0, 2, 4, 6, . . . }

O = { 1, 3, 5, 7, . . . }

The Set of Odd Numbers

The Set of Even Numbers

The Set of Natural Numbers or Counting Numbers

Sets N, E and O are all proper subsets of Set W

The Sets of Whole Numbers (W), Natural Numbers (N), Even Numbers

(E), and Odd Numbers (O) are all infinite sets

The elements of any of these sets can be matched in a one-to-one

correspondence with the elements of any other of these sets.

Unlike a finite set, an infinite set can have a one-to-one correspondence with one of

its proper subsetsIn fact, the definition of an infinite set is a

set that can be put in a one-to-one correspondence with a proper subset of

itself

Finding All the Subsets of a Finite Set of Whole Numbers

Example: What are the subsets of set A = {a, b, c} ?

{ }, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, and {a,b,c}Every set has the empty set as well as the entire set in their list

of subsetsThe number of subsets of a finite set = 2n,

where n equals the number of elements in the finite

set.Example: What are the number of subsets for set A ?

23 = 8 subsets for set A

Section 2.1

Linda Roper

The End