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Welcome to MM150 – Unit 2 Seminar

Unit 2 Seminar
Instructor: Larry Musolino
Email: [email protected]
2.1 – Set Concepts
2.2 – Subsets
2.3 –Venn Diagrams and Set Operations
2.4 – More Venn Diagrams
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 1
Reminder on Seminars

Reminder on Seminars:
 Seminars Available (attend any one):
Wednesday
Unit 1
Anne Poe Weds
June 6th 12noon
Thursday
Charlie Johnson Thursday
June 7th 10 PM
Unit 4
Dawn Thomas Weds Charlie Johnson Thursday
June 13th 12noon
June 14th 10 PM
Anne Poe Weds
June 20th 12noon
Anne Poe Weds
June 27th 12noon
Unit 5
Charlie Johnson Thursday
July 5th 10 PM
Unit 2
Unit 3
Unit 6
Dawn Thomas Weds
July 11th 12noon
Unit 7
Dawn Thomas Weds
July 18th 12noon
Unit 8
Dawn Thomas Weds
July 25th 12noon
Anne Poe Weds
Aug 1st 12noon
Times are Eastern Time Zone
Unit 9
Copyright © 2009 Pearson Education, Inc.
Saturday
Larry Musolino Saturday
June 16th 11am
Larry Musolino Saturday
June 30th 11am
Larry Musolino Saturday
July 14th 11am
Larry Musolino Saturday
July 28th 11am
Charlie Johnson Thursday
Aug 2nd 10 PM
Slide 2 - 2
Weekly Responsibilities


Readings and Video Lectures
Attend Weekly Seminar



Optional, attendance is not graded.
Participate in weekly discussion topic, note you
must post initial response and then at least two
follow-up responses to other students
Complete the MyMathLab (MML) Graded
Practice (20 problems).

Reminder on MML Graded Practice:


Must be completed by Tues 11:59ET for each weekly Unit.
If you have questions or encounter problems with MML, post
your questions in Discussion Board, or contact Math Center
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 3
Summary of Student Responsibilities
Student Responsibilities
Due Date
For weekly Units 1 to 9 (DISCUSSION BOARD):
(a) Provide initial response to the Discussion Board
Question
(b) Post at least 2 follow-up responses to fellow students
By the end of the
weekly unit, Tuesdays
11:59pm ET
For weekly Units 1 to 9 (SEMINAR):
(a) Attend one of the Live Seminars as shown on previous
Slide
(b) Attending the live seminar is OPTIONAL and you are
not graded on SEMINAR attendance
Live Seminar times as
shown on previous
slide.
For weekly Units 1 to 9 (MML GRADED PRACTICE):
(a) Complete the MyMathLab set of 20 problems
By the end of the
weekly unit, Tuesdays
11:59pm ET
For Unit 9
(FINAL PROJECT):
(a) Post the Final Project in the Course
Final Project due by end
of week 9
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 4
Accessing Kaplan Math Center


Live, one-on-one tutoring.
To access the Math Center:
For live tutoring: go to “My Studies” on the KU Campus
 choose “Academic Support Center”
 select “Math Tutoring
In the past students have indicated this is a great resource, please take
advantage of this !!!!


Copyright © 2009 Pearson Education, Inc.
Slide 2 - 5
2.1
Set Concepts
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Slide 2 - 6
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.

The symbol  read “is an element of,” is used
to indicate membership in a set.

The symbol
 means “is not an element of.”
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 7
Sets - Applications

Many applications of set and set operations in real-life:

Biology – taxonomy is science of classifying living things in sets,



Business Application – Insurance companies pool together
applicants in sets based on various risk factors such as age,
gender, smoker vs. non-smoker, etc. for actuarial analysis.
Scientific studies – During clinical trials, patients are arranged in
various sets for evaluation:


Example: Living organisms classified into six sets called: animalia,
plantae, archaea, eubacteria, fungi, protista.
E.g. placebo vs. non-placebo
Chemistry – Elements are classified into certain sets in the
Periodic Table with common properties, e.g metals, non-metals,
noble gases, etc.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 8
Well-defined Set

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.

Examples:

Well-Defined Set:



Set of Integers between 4 and 9 inclusive:
{4, 5, 6, 7, 8, 9}
Not Well-Defined Set:


Set of Three Best NFL Teams of All Time.
Subjective and Set Elements will vary depending on opinions.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 9
Roster Form

This is the form of the set where the elements are all
listed within curly braces { } , 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.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 10
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 setbuilder notation.
Solution:


B  x x N and x is an even number  10 .
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 11
You Try It #1

Write the following set in both set-builder
notation and roster notation:

Set A is the set of natural numbers between 4
and 10, inclusive.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 12
You Try It #1 - Solution

Write the following set in both set-builder notation and
roster notation:

Set A is the set of natural numbers between 4 and 10,
inclusive.

Solution:

Roster Notation: A = { 4, 5, 6, 7, 8, 9, 10 }

Set-Builder Notation:
Copyright © 2009 Pearson Education, Inc.
A  {x | x  N and 4  x  10}
Slide 2 - 13
You Try It #2

Write the following set in both set-builder
notation and roster notation:

Set B is the set of natural numbers greater than
12.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 14
You Try It #2 - Solution

Write the following set in both set-builder notation and
roster notation:

Set B is the set of natural numbers greater than 12.

Solution:

Roster Notation: B = { 13, 14, 15, 16, … }

Set-Builder Notation:
Copyright © 2009 Pearson Education, Inc.
B  {x | x  N and x  12}
Slide 2 - 15
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 - 16
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.
N = {1, 2, 3, 4, 5, …}
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 17
Cardinal Number

The number of elements in set A is its cardinal
number.

Symbol: n(A)

Example:


Set A = {Toyota, Ford, Chevy, Honda}
n(A) = 4
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Slide 2 - 18
Equal Sets

Equal sets have the exact same elements in
them, regardless of their order.

Symbol:

Example of Equal Sets:


A=B
Set A = {Toyota, Ford, Chevy, Honda}
Set B = {Chevy, Honda, Ford, Toyota}
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Slide 2 - 19
Equivalent Sets

Equivalent sets have the same number of elements in
them.
Symbol: n(A) = n(B)

Example of Equivalent Sets:





Set A = {Toyota, Ford, Chevy, Honda}
Set B = {Dolphins, Jets, Packers, Eagles}
Note that if Two Sets are Equal then they must also be
equivalent.
Note that if Two Sets are Equivalent they are not
necessarily equal.
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Slide 2 - 20
Empty (or Null) Set

A null set (or empty set ) contains absolutely
NO elements.

Symbol:
 or
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 
Slide 2 - 21
Universal Set

The universal set contains all of the possible
elements which could be discussed in a
particular problem.

Symbol: U
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Slide 2 - 22
You Try It #3

Given that:
C  {x | x  N and 2  x  8}
D  {x | x  6  10}
E { }

Find n(C), n(D), n(E)
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 23
You Try It #3 - Solution

Given that:
C  {x | x  N and 2  x  8}
D  {x | x  6  10}
E { }
Find n(C), n(D), n(E)



n(C) = 6
n(D) = 1
n(E) = 0
since C = { 2, 3, 4, 5, 6, 7 }
since D = { 4 }
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 24
You Try It #4


State whether the two Sets are Equal, Equivalent, Both,
or Neither:
Page 76 Problem 79



Page 76 Problem 81



Set A = {algebra, geometry, trigonometry}
Set B = {geometry, trigonometry, algebra}
Set A = {grapes, apples, oranges}
Set B = {grapes, peaches, apples, oranges}
Page 76 Problem 83


Set A is the set of letters in word tap
Set B is the set of letters in word ant
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 25
You Try It #4 - Solution


State whether the two Sets are Equal, Equivalent, Both, or
Neither:
Page 76 Problem 79



Answer: Both
Page 76 Problem 81



Set A = {algebra, geometry, trigonometry}
Set B = {geometry, trigonometry, algebra}
Set A = {grapes, apples, oranges}
Set B = {grapes, peaches, apples, oranges}
Answer: Neither
Page 76 Problem 83

Set A is the set of letters in word tap
Answer:
Equivalent,

Set B is the set of letters in word ant
Not Equal
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Slide 2 - 26
2.2
Subsets
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Slide 2 - 27
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: A  B, means A is a subset of B.
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. The symbol for “not a subset of” is 
Notice that for any Set A, A  A
 That is, every set is also a subset of itself.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 28
Determining Subsets
Example:
Determine whether set A is a subset of set B.
Also determine whether set A is subset of set C.
A = { 3, 5, 6, 8 }
B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C = {1, 2, 3, 4, 5, 6, 7}
Solution:
All of the elements of set A are contained in set
B, so: A  B.
All of the elements of set A are not contained in
set C, so: A C .
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 29
Proper Subset




All subsets are proper subsets except the subset
containing all of the given elements, that is, the given
set must contain one element not in the subset (the
two sets cannot be equal).
Written Another Way:
 Set A is proper subset of Set B if and only if all
elements of set A are elements of set B and
set A  B
Symbol for Proper Subset:
Symbol for Not Proper Subset:
Copyright © 2009 Pearson Education, Inc.
A B
A B
Slide 2 - 30
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.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 31
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.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 32
You Try It #5
Determine whether set A is a proper subset of
set B.
A = { x | x  N and x < 6 }
B = { x | x  N and 1  x  5 }
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 33
You Try It #5 - Solution
Determine whether set A is a proper subset of
set B.
A = { x | x  N and x < 6 }
B = { x | x  N and 1  x  5 }
Solution:
All the elements of set A are contained in set B,
but sets A and B are equal, therefore A  B,
that is, set A is not a proper subset of set B.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 34
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 }.

Copyright © 2009 Pearson Education, Inc.
Slide 2 - 35
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}, { }
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 36
You Try It #6

Given that:
C  {x | x  N and 2  x  8}
D  {x | x  6  10}
E { }
(a)
(b)
How many distinct subsets can be made from Sets C, D, E?
How many distinct proper subsets can be made from Sets C,
D, E?
Reminder: The number of distinct subsets of a set A is 2n,
where n is the number of elements in set A.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 37
You Try It #6 - Solutions

Given that:
C  {x | x  N and 2  x  8}
D  {x | x  6  10}
E { }
(a)
How many distinct subsets can be made from Sets C, D, E?



(b)
Set C: Number Distinct Subsets = 26 = 64
Set D: Number Distinct Subsets = 21 = 2
Set E: Number Distinct Subsets = 20 = 1
How many distinct proper subsets can be made from Sets C, D, E?




Set C: Number Distinct Proper Subsets = 26 – 1 = 63
Set D: Number Distinct Proper Subsets = 21 – 1 = 1
Set E: Number Distinct Proper Subsets = 20 – 1 = 0
Based on this you can see that the Number of Distinct Proper
Subsets for any set is 2n – 1 where n is number elements.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 38
2.3
Venn Diagrams and Set
Operations
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Slide 2 - 39
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 may be divided
into subsets of U and are represented by
circles.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 40
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
are no elements in common
since there is no overlapping area of the two circles.
U
A
Copyright © 2009 Pearson Education, Inc.
B
Slide 2 - 41
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 that are in the
intersection of set A and set
B.
Elements in the intersection
are elements that are in
common for Sets A and B.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 42
Setting up Venn Diagram with Two
Overlapping Sets (See page 85)
Notice Region I consists of elements which appear only in SET A.
Notice Region III consists of elements which appear only in SET B.
Notice Region II (Middle Region) consists of elements which appear in
both Set A and Set B.
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Slide 2 - 43
Example of Venn Diagram for Two Overlapping Sets

To fill in a Venn Diagram, Start with the Middle
Region and then work outward.
Students
Taking
Math
Class
Students
Taking
History
Class
James
Alice
Mike
James
Alice
Harry
Bob
Jen
A
Set of students
taking math
only
Mike
Bob
Copyright © 2009 Pearson Education, Inc.
James
Alice
B
Set of students
taking both
math and
history
Set of students
taking history
only
Harry
Jen
Slide 2 - 44
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’
Stated another way:
The complement of Set A is all of the elements in U which are not in A
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Slide 2 - 45
Intersection

The intersection of two given sets contains only
those elements common to both of those sets.

Symbol:
AB
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Slide 2 - 46
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 - 47
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 - 48
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.
Copyright © 2009 Pearson Education, Inc.
U
A
B
Slide 2 - 49
The Meaning of and and or

and is generally interpreted to mean intersection

or is generally interpreted to mean union
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Slide 2 - 50
The Relationship Between n(A  B),
n(A), n(B), n(A  B)

To find the number of elements in the union of
two sets A and B, we add the number of
elements in set A and B and then subtract the
number of elements common to both sets.
n(A B) = n(A) + n(B) – n(A  B)
n(A or B) = n(A) + n(B) – n(A and B)
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 51
You Try It #7

Given following sets:





U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 4, 6 }
B = { 1, 3, 6, 7, 9 }
C={ }
Find the following:






AB
AC
AB
A’  B
(A B)’
n(A), n(B), n(A  B), n(A  B)
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 52
You Try It #7 - Solutions

Given following sets:





U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 4, 6 }
B = { 1, 3, 6, 7, 9 }
C={ }
Find the following:







A  B = { 1, 2, 3, 4, 6, 7, 9 }
A  C = { 1, 2, 4, 6 } (Note that A  C = A)
A  B = { 1, 6 }
A’  B = { 3, 5, 7, 8, 9, 10 }  { 1, 3, 6, 7, 9 }
= { 1, 3, 5, 6, 7, 8, 9, 10 }
(A  B)’ = { 5, 8, 10 }
n(A) = 4, n(B) = 5, n(A  B) = 2, n(A  B) = 7
Notice that n(A B) = n(A) + n(B) – n(A  B)
= 4 + 5 2 =7
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 53
You Try It #7 - Solutions

Given following sets:





U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 4, 6 }
B = { 1, 3, 6, 7, 9 }
C={ }
Find the following:







Solution Using Venn Diagram:
2
4
A  B = { 1, 2, 3, 4, 6, 7, 9 }
A  C = { 1, 2, 4, 6 }
A  B = { 1, 6 }
A’  B = { 3, 5, 7, 8, 9, 10 }  { 1, 3, 6, 7, 9 }
= { 1, 3, 5, 6, 7, 8, 9, 10 }
(A  B)’ = { 5, 8, 10 }
n(A) = 4, n(B) = 5, n(A  B) = 2, n(A  B) = 7
Notice that n(A B) = n(A) + n(B) – n(A  B)
= 4 + 5 2 =7
Copyright © 2009 Pearson Education, Inc.
1
6
3
7
9
5 8 10
Slide 2 - 54
Difference of Two Sets

The difference of two sets A and B symbolized
A – B, is the set of elements that belong to set
A but not to set B. Region 1 represents the
difference of the two sets.

A  B  x | x A and x B

U
A
B
I
II III
IV
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 55
Cartesian Product

The Cartesian product of set A and set B,
symbolized A  B, and read “A cross B,” is
the set of all possible ordered pairs of the
form (a, b), where a  A and b  B.

Select the first element of set A and form an
ordered pair with each element of set B. Then
select the second element of set A and form
an ordered pair with each element of set B.
Continue in this manner until you have used
each element in set A.
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Slide 2 - 56
2.4
Venn Diagrams with Three Sets
And
Verification of Equality of Sets
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Slide 2 - 57
General Procedure for Constructing Venn
Diagrams with Three Sets

Determine the
elements that are
common to all three
sets and place in
region V, A  B  C.
U
II
I
A
III
V
IV
VI
B
VII
C
Copyright © 2009 Pearson Education, Inc.
VIII
Slide 2 - 58
General Procedure for Constructing Venn
Diagrams with Three Sets continued




Determine 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.
Copyright © 2009 Pearson Education, Inc.
U
II
I
A
III
V
IV
VI
B
VII
C
VIII
Slide 2 - 59
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.
Copyright © 2009 Pearson Education, Inc.
U
II
I
A
III
V
IV
VI
B
VII
C
VIII
Slide 2 - 60
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.
Copyright © 2009 Pearson Education, Inc.
U
II
I
A
III
V
IV
VI
B
VII
C
VIII
Slide 2 - 61
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 }
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 62
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 IV.
A  C = {1, 5, 8}
Since 5 has been placed in region V, place 1
and 8 in region IV.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 63
Example: Constructing a Venn diagram
for Three Sets continued


Now determine the numbers that go in region
VI. B  C = {5}
There are no new numbers to be placed in
region VI.
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, in
which we must write 4. Using set C completes
region VII, in which we must write 3.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 64
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
Copyright © 2009 Pearson Education, Inc.
7
C
VIII
Slide 2 - 65
Verification of Equality of Sets

To verify set statements are equal for any two
sets selected, we use deductive reasoning
with Venn Diagrams.

If both statements represent the same regions
of the Venn Diagram, then the statements are
equal for all sets A and B.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 66
Example: Equality of Sets
Determine whether (A  B)’= A’ B’ for all sets
A and B.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 67
Solution
U
A
Draw a Venn diagram
with two sets A and B
and label each region.
Find (A  B)’
B
I
II III
IV
Find A’  B’
Set
Regions
Set
Regions
A
I, II
A’
III, IV
B
II, III
B’
I, IV
II
A’ B’
I, III, IV
AB
(A  B)’
I, III, IV
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 68
Solution
Set
Regions
Set
Regions
A
I, II
A’
III, IV
B
II, III
B’
I, IV
II
A’ B’
I, III, IV
AB
(A  B)’
I, III, IV
Both statements are represented by the same
regions, I, III, and IV, of the Venn diagram. Thus,
(A  B)’ = A’  B’ for all sets A and B.
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 69
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  B)’ = A’ B’
(A  B)’ = A’  B’
Copyright © 2009 Pearson Education, Inc.
Slide 2 - 70