Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 2.1 Set Concepts 2.1-1.
Section 2.1
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Transcript of Section 2.1
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