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Sets and Whole Numbers

Mathematics for Elementary School Teachers - 4th Edition

O’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?

Blue

a

C =

A =

e

Red

o

i

Yellow

Sets and Whole Numbers - Section 2.1

A set is a collection of objects

or ideas that can be listed or described

A set is usually listed with a capital letter

A set can be represented using braces { }

A = {a, e, i, o, u}

C = {Blue, Red, Yellow}

A set can also be represented using a circle

C =

Red

Yellow

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

Blue is an element of set C

Orange is not an element of set C

Set B

1

a

2

b

c

3

Definition of a One-to-One Correspondence

Sets A and B have a one-to-onecorrespondence 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.

The order of the elements does not matter

Sets 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 = B

So equal sets are when both sets contain the same elements - but the order of the elements does not matter

~

B

Set B

Set A

Dog

one

two

Cat

three

Frog

Definition of Equivalent Sets

Sets A and B are equivalent sets if and only if there is a one-to-one correspondence between A and B

Whole Numbers

Set A

Natural

Numbers

Definition of a Subset of a Set

For 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

Example: The Natural Numbers and the Whole Numbers

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

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

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

A⊈B

Example:

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

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

set A contains animals that are not in set B

Thus, A⊈B

Whole Numbers

Set A

Natural

Numbers

Definition of a Proper Subset of a Set

For 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.

A ⊂ B

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

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

N ⊂ W

Red

Blue

Yellow

The Universal Set, U

The 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

A set with no elements

Symbols for the empty set: { } or ∅

Complement of a set

The complement of a set A is all the elements in the universal set that are not in A

A set with an unlimited number of elements

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

Finite Set

A set with a limited number of elements

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

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) = 4

So, 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 is the process that enables people systematically to associate a whole number with a set of objects.” (class text, p. 65)

“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)

A = {Dog, Cat, Frog, Mouse}

B = { 1, 2, 3, 4 }

For 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:

A = {dog, cat, fox, rabbit}

n(A) = 4

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

n(B) = 5

4 is less than 5 (4 < 5)

Set A is equivalent to a proper subset of B

The Set of Odd Numbers

The Set of Natural Numbers or Counting Numbers

The Set of Whole Numbers

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

Important Subsets of the Whole Numbers

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

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

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

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 subsets

In 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 subsets

The 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

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