Transcript Section 2.1

Section 2.1
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?
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
A=
a u
e o
i
C=
Blue
Red
Yellow
Each object in the set is called an element of the set
C=
Blue
Red
Yellow
Blue is an element of set C
Blue  C
Orange is not an element of set C
Orange  C

Definition of a One-to-One Correspondence
Sets 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
Set B
1
a
2
b
3
c
The order of the elements does not matter
Definition of Equal Sets
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
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
Set A
Set B
one
two
three
Dog
Cat
Frog
A~B
Definition of a Subset of a Set
Set B
For all sets A and B, A is a subset of B if and
Set A
only if each element of A is also an element
A B
of B
Example: The Natural Numbers and the Whole
Numbers
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
W
=
{0,
1,
2,
3,
4,
5,
.
.
.
}

12 . . . }
Whole Numbers
Natural
Numbers
N W
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
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.
Set B
A⊂ B
Set A
Whole Numbers
Natural
Numbers
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12 . . . }
W = {0, 1, 2, 3, 4, 5, . . . }
N⊂ W
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
U
Red
Blue
Yellow
The Empty Set
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
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) = 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 and Sets
“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 }
Less Than and Greater Than
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 Whole Numbers
W = { 0, 1, 2, 3, 4, . . . }
Important Subsets of the Whole Numbers
The Set of Natural Numbers or Counting Numbers
N = { 1, 2, 3, 4, . . . }
The Set of Even Numbers
E = { 0, 2, 4, 6, . . . }
The Set of Odd Numbers
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
n, where n
The number of subsets of a finite set = 2
equals the number of elements in the finite set.
Example: What are the number of subsets for
set A ?
3 = 8 subsets for set A
2
The End
Section 2.1
Linda Roper