Transcript aar07_02

Copyright © 2005 Pearson Education, Inc.
SEVENTH EDITION and EXPANDED SEVENTH EDITION
Slide 2-1
Chapter 2
Sets
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2.1
Set Concepts
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Set

A collection of objects, which are called
elements or members of the set.

Listing the elements of a set inside a pair of
braces, { }, is called roster form .
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Slide 2-4
Well-defined Set



A set which has no question about what
elements should be included.
Its elements can be clearly determined.
No opinion is associated with the members.
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Slide 2-5
Roster Form

This is the form of the set where the elements are all
listed, each separated by commas.
Example:
Set N is the set of all natural numbers less than or equal
to 25.
Solution: N = {1, 2, 3, 4, 5,…25}
The 25 after the ellipsis indicates that the elements
continue up to and including the number 25.
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Slide 2-6
Set-Builder (or Set-Generator) Notation
A formal statement that describes the members of a set
is written between the braces.
 A variable may represent any one of the members of
the set.
Example: Write set B = {2, 4, 6, 8, 10} in set-builder
notation.
Solution:

B  { x x  N and x is an even number  10}.
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Slide 2-7
Finite Set
A set that contains no elements or the number
of elements in the set is a natural number.
Example:
Set S = {2, 3, 4, 5, 6, 7} is a finite set because
the number of elements in the set is 6, and 6 is
a natural number.

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Slide 2-8
Infinite Set


An infinite set contains an indefinite
(uncountable) number of elements.
The set of natural numbers is an example of an
infinite set because it continues to increase
forever without stopping, making it impossible to
count its members.
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Slide 2-9
Equal Sets

Equal sets have the exact same elements in
them, regardless of their order.

Symbol:
A=B
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Slide 2-10
Cardinal Number

The number of elements in set A is its cardinal
number.

Symbol: n(A)
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Slide 2-11
Equivalent Sets

Equivalent sets have the same number of
elements in them.

Symbol: n(A) = n(B)
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Slide 2-12
Empty (or Null) Set

A null (or empty set ) contains absolutely NO
elements.

Symbol:
 or
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 
Slide 2-13
Universal Set

The universal set contains all of the possible
elements which could be discusses in a
particular problem.

Symbol: U
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Slide 2-14
2.2
Subsets
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Subsets

A set is a subset of a given set if and only if all
elements of the subset are also elements of
the given set.

Symbol:

To show that set A is not a subset of set B, one
must find at least one element of set A that is
not an element of set B.

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Slide 2-16
Determining Subsets
Example:
Determine whether set A is a subset of set B.
A = { 3, 5, 6, 8 }
B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Solution:
All of the elements of set A are contained in set
B, so A  B.
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Slide 2-17
Proper Subset

All subsets are proper subsets except
the subset containing all of the given
elements.

Symbol:

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Slide 2-18
Determining Proper Subsets
Example:
Determine whether set A is a proper subset of set B.
A = { dog, cat }
B = { dog, cat, bird, fish }
Solution:
All the elements of set A are contained in set B, and sets
A and B are not equal, therefore A  B.
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Slide 2-19
Determining Proper Subsets continued
Example:
Determine whether set A is a proper subset of set B.
A = { dog, bird, fish, cat }
B = { dog, cat, bird, fish }
Solution:
All the elements of set A are contained in set B, but sets
A and B are equal, therefore A  B.
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Slide 2-20
Number of Distinct Subsets
The number of distinct subsets of a finite set A
is 2n, where n is the number of elements in set
A.
Example:

Determine the number of distinct subsets for
the given set { t , a , p , e }.

List all the distinct subsets for the given set:
{ t , a , p , e }.

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Slide 2-21
Number of Distinct Subsets continued
Solution:

Since there are 4 elements in the given set, the
number of distinct subsets is
24 = 2 • 2 • 2 • 2 = 16 subsets.

{t,a,p,e},
{t,a,p}, {t,a,e}, {t,p,e}, {a,p,e},
{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e},
{t}, {a}, {p}, {e}, { }
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Slide 2-22
2.3
Venn Diagrams and
Operations
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Set
Venn Diagrams


A Venn diagram is a technique used for
picturing set relationships.
A rectangle usually represents the universal set,
U.

The items inside the rectangle are divided into
subsets of U and are represented by circles.
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Slide 2-24
Disjoint Sets



Two sets which have no elements in common are said
to be disjoint.
The intersection of disjoint sets is the empty set.
Disjoint sets A and B are
drawn in this figure. There
U
are no elements in common
since there is no overlapping area of the two circles.
A
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B
Slide 2-25
Overlapping Sets


For sets A and B drawn
in this figure, notice the
overlapping area shared
by the two circles.
This section represents
the elements are in the
intersection of set A and
set B.
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U
A
B
Slide 2-26
Complement of a Set

The set known as the complement contains all
the elements of the universal set, which are not
listed in the given subset.

Symbol: A’
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Slide 2-27
Intersection

The intersection of two given sets contains only
those elements common to those sets.

Symbol: A
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B
Slide 2-28
Union



The union of two given sets contains all of the
elements for those sets.
The union “unites” that is, it brings together
everything into one set.
Symbol: A
B
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Slide 2-29
Subsets


When B  A, every
element of B is also
an element of A.
Circle B is completely
inside circle A.
U
B
A
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Slide 2-30
Equal Sets


When set A is equal
to set B, all the
elements of A are
elements of B, and
all the elements of B
are elements of A.
Both sets are drawn
as one circle.
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U
A
B
Slide 2-31
2.4
Venn Diagrams with
Three Sets
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General Procedure for Constructing Venn
Diagrams with Three Sets

Find the elements
that are common to
all three sets and
place in region V.
U
II
I
A
III
V
IV
VI
B
VII
C
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VIII
Slide 2-33
General Procedure for Constructing Venn
Diagrams with Three Sets continued

Find the elements for
region II. Find the
elements in A B . The
elements in this set
belong in regions II and
V. Place the elements
in the set A B that are
not listed in region V in
region II. The elements
in regions IV and VI are
found in a similar
manner.
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U
II
I
A
III
V
IV
VI
B
VII
C
VIII
Slide 2-34
General Procedure for Constructing Venn
Diagrams with Three Sets continued

Determine the
elements to be placed
in region I by
determining the
elements in set A that
are not in regions II, IV,
and V. The elements in
regions III and VII are
found in a similar
manner.
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U
II
I
A
III
V
IV
VI
B
VII
C
VIII
Slide 2-35
General Procedure for Constructing Venn
Diagrams with Three Sets continued

Determine the
elements to be placed
in region VIII by finding
the elements in the
universal set that are
not in regions I
through VII.
U
II
I
A
V
IV
VI
B
VII
C
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III
VIII
Slide 2-36
Example: Constructing a Venn diagram
for Three Sets
Construct a Venn diagram illustrating the following sets.
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = { 1, 2, 5, 8}
B = {2, 4, 5}
C = {1, 3, 5, 8}
Solution:

Find the intersection of all three sets and place in
region V, A B C  {5}.
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Slide 2-37
Example: Constructing a Venn diagram
for Three Sets continued


Determine the intersection of sets A and B
and place in region II.
A B  {2, 5}
Element 5 has already been placed in region V, so 2
must be placed in region II.
Now determine the numbers that go into region V.
A C  { 1, 2, 5, 8}
Since 5 has been placed in region V, place 1 and 8 in
region IV.
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Slide 2-38
Example: Constructing a Venn diagram
for Three Sets continued

Now determine the numbers that go in region
VI. B C  {5}

There are now new numbers to be placed in this
region. Since all numbers in set A have been placed,
there are no numbers in region I. The same
procedures using set B completes region III. Using set
C completes region VII.
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Slide 2-39
Example: Constructing a Venn diagram
for Three Sets continued

The Venn diagram is then completed.
U
4
2
II
I
A
III
5
1,8 V
IV
VI
B
3
VII
6
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7
C
VIII
Slide 2-40
De Morgan’s Laws


A pair of related theorems known as
De Morgan’s laws make it possible to change
statements and formulas into more convenient
forms.
(A
(A
B) = A 
B) = A 
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B
B
Slide 2-41
2.5
Applications of Sets
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Example: Toothpaste Taste Test





A drug company is considering manufacturing a new
toothpaste. They are considering two flavors, regular
and mint.
In a sample of 120 people, it was found that 74 liked
the regular, 62 liked the mint, and 35 liked both types.
How many liked only the regular flavor?
How many liked either one or the other or both?
How many people did not like either flavor?
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Slide 2-43
Solution


Begin by setting up a
Venn diagram with
sets A (regular flavor)
and B (mint flavor).
Since some people
liked both flavors, the
sets will overlap and
the number who liked
both with be placed in
region II.
35 people liked both
flavors.
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U
35
Regular
Mint
Slide 2-44
Solution continued


Next, region I will refer
to those who liked only
the regular and region III
will refer to those who
liked only the mint.
In order to get the
number of people in
each region, find the
difference between all
the people who liked
each toothpaste and
those who liked both.
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74 – 35 = 39
62 – 35 = 27

U
39
regular
only
Regular
35
both
27
mint
only
Mint
Slide 2-45
Solution continued

“One or the other or both” represents the UNION of
the two sets.

Therefore, 39 + 27 + 35 = 101 people who liked
one or the other or both.
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Slide 2-46
Solution continued

Take the total number of people in the entire sample
and subtract the number who liked one or the other or
both.
19 people did not like either flavor.
19 liked
neither
U
74-35=39
Liked mint
only
Regular
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35
both
62-35=27
Liked mint
only
Mint
Slide 2-47
2.6
Infinite Sets
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Infinite Sets

An infinite set is a set that can be placed in a
one-to-one correspondence with a proper
subset of itself.

These sets are “unbounded”.
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Slide 2-49
Example: The Set of Multiples of Four
Show that it is an infinite set.
{4, 8, 12, 16, 20, …,4n, …}
Solution: We establish one-to-one
correspondence between the counting numbers
and a proper subset of itself.
Given set:
{4, 8, 12, 16, 20, …, 4n, …}
Proper subset: {4, 8, 12, 16, 20, …, 4n + 4, …}
Therefore, the given set is infinite.
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Slide 2-50
Countable Sets


A set is countable if it is finite or if it can be
placed in a one-to-one correspondence with the
set of counting numbers.
Any set that can be placed in a one-to-one
correspondence with a set of counting numbers
has cardinality aleph-null and is countable.
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Slide 2-51