Set Theory
Symbols and Terminology
Set – A collection of objects
Set Theory
Symbols and Terminology
Element – An object in a set
Set Theory
Symbols and Terminology
Empty (Null) Set – A set that
contains no elements
Set Theory
Symbols and Terminology
Cardinal Number
(Cardinality) – The number of
elements in a set
Set Theory
Symbols and Terminology
Finite Set – A set that contains a
limited number of elements
Set Theory
Symbols and Terminology
Infinite Set – A set that contains an
unlimited number of elements
Set Theory
Symbols and Terminology
There are three ways to describe a set
Word Description
Listing
Set Builder Notation
Set Theory
Symbols and Terminology
The following example shows the three
ways we can describe the same set.
Set Theory
Symbols and Terminology
Word Description
“The set of even counting numbers less
than 10”
Set Theory
Symbols and Terminology
Listing
E = {2, 4, 6, 8 }
Set Theory
Symbols and Terminology
Set Builder Notation
E = {x | x is an even counting number
that is less than 10 }
Set Theory
Symbols and Terminology
Cardinal Numbers
n(E) means “the number of elements in
set E”
In this particular case n(E) = 4
Example 1) Suppose A is the set of
all lower case letters of the alphabet.
We could write out set A as follows:
A = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r
s,t,u,v,w,x,y,z}
We can shorten this notation if we
clearly show a pattern, as in the
following:
A = {a,b,c,d,e…,w,x,y,z}
Try writing the following three sets
by listing the elements.
1) The set of counting numbers
between six and thirteen.
1)The set of counting numbers
between six and thirteen.
B = {7, 8, 9, 10, 11, 12 }
2) C = {5, 6, 7, …, 13}
2) C = {5, 6, 7, …, 13}
C = {5, 6, 7, 8, 9, 10, 11, 12, 13}
3) D = {x | x is a counting number
between 6 and 7}
3) D = {x | x is a counting number
between 6 and 7}
D = { } or Ø
Homework
• Page 54 # 1 - 10
Warm Up
Find the cardinal number of each set
K = {2, 4, 8, 16}
M={0}
R = {4, 5, …, 12, 13}
P=Ø
Warm Up
Find the cardinal number of each set
K = {2, 4, 8, 16} n(K)=4
M = { 0 } n(M)=1
R = {4, 5, …, 12, 13} n(R)=10
P = Ø n(P)=0
Pg 54 #1-10 Answers
1. C
5. B
2. G
6. D
3. E
7. H
4. A
8. F
9. A = {1, 2, 3, 4, 5, 6}
10. B = {9, 10, 11, 12, 13, 14, 15, 16, 17}
Set Theory
Symbols and Terminology
Empty Set – Example)
P = {x | x is a positive number <0}
Therefore P = { } or P = Ø but
P ≠ {Ø}
Set Theory
Symbols and Terminology
Infinite Set – Example)
R = {y | y is an odd whole number}
Therefore R = {1, 3, 5, 7, …}
Set Theory
Symbols and Terminology
Finite Set – Example)
F = {z | z is a factor of 30}
Therefore F = {1, 2, 3, 5, 6, 10, 15, 30}
Classwork
Page 54 & 55 #11 – 49 odd
Page 54 & 55 #11 – 49 odd
11. The set of all whole numbers not
greater than 4 can be expressed by listing
as
A ={0, 1,2,3, 4}.
13. In the set {6, 7,8.... , 14}, the ellipsis
(three dots) indicates a continuation of
the pattern. A complete listing oft his set
is B ={6,7,8,9,10, 11, 12, 13,14}.
Page 54 & 55 #11 – 49 odd
15. The set { -15, -13, 11,..., -1} contains all
integers from -15 to -1 inclusive. Each member
is two larger than its predecessor. A complete
listing of this set is
C ={- 15, -13, -11, -9, -7, -5, -3, -1}.
17. The set {2, 4, 8, ... , 256} contains all powers
of two from 2 to 256 inclusive. A complete
listing of this set
D={2, 4,8,16,32,64,128, 256}.
Page 54 & 55 #11 – 49 odd
19. A complete listing of the set {x x is an even
whole number less than 11 } is
E={0, 2, 4, 6, 8, 10}. Remember that 0 is the first
whole number.
21. The set of all counting numbers greater than 20
is represented by the listing F={21, 22, 23,... }.
Page 54 & 55 #11 – 49 odd
23. The set of Great Lakes is represented by
G={Lake Erie, Lake Huron, Lake Michigan, Lake
Ontario, Lake Superior}.
25. The set {x | x is a positive multiple of 5} is
represented by the listing H={5, 10,15,20,.,. }.
Page 54 & 55 #11 – 49 odd
27. The set {x|x is the reciprocal of a natural
number} is represented by the listing
I={1, 1/2, 1/3, 1/4, 1/5, ... }.
29. The set of all rational numbers may be
represented using set-builder notation as
J={x|x is a rational number}.
Page 54 & 55 #11 – 49 odd
31. The set {1, 3,5,... , 75} may be represented
using set- builder notation as
K={x|x is an odd natural number less than
76}.
33. The set {2, 4, 6,... , 32} is finite since the
cardinal number associated with this set is a
whole number.
Page 54 & 55 #11 – 49 odd
35. The set {112, 2/3, 3/4, ... } is infinite since there
is no last element, and we would be unable to
count all of the elements.
37. The set {x|x is a natural number greater than 50}
is infinite since there is no last element, and
therefore its cardinal number is not a whole
number.
Page 54 & 55 #11 – 49 odd
39. The set {x|x is a rational number} is infinite
since there is no last element, and therefore its
cardinal number is not a whole number;
41. For any set A, n(A) represents the cardinal
number of the set, that is, the number of elements
in the set. The set A = {0, 1, 2, 3, 4, 5, 6, 7}
contains 8 elements. Thus, n(A) = 8.
Page 54 & 55 #11 – 49 odd
43. The set A = {2, 4, 6, ... , l000} contains
500 elements. Thus, n(A) = 500.
45. The set A = {a, b, c, ,.. , z} has 26
elements (letters of the alphabet). Thus
n(A) = 26.
Page 54 & 55 #11 – 49 odd
47. The set A = the set of integers between -20
and 20 has 39 members. The set can be
indicated as {- 19, -18,...,18, 19}, or 19
negative integers, 19 positive integers, and
0. Thus, n(A) = 39.
49. The set A = { 1/3, 2/4, 3/5, 4/6, ..., 27/29,
28/30} has 28 elements. Thus, n(A) = 28.
Equal and Equivalent Sets
Equal Sets – Two sets are equal if they contain
the EXACT same elements.
A={1,4,9,16,25}
B={1,9,4,25,16}
Equal and Equivalent Sets
Equivalent Sets – Two sets are equivalent if
they contain the same NUMBER of
elements.
A={1,3,5,7,9}
B={1,2,4,8,16}
Well Defined and Not Well Defined
Sets
Well Defined
A  {x | x is a prime number}
Not Well Defined
A  {x | x is a pretty number}
Well Defined and Not Well Defined
Sets
Well Defined
B  {x | x is a student in this class}
Not Well Defined
B  {x | x is a good student in this class}
Well Defined and Not Well Defined
Sets
Well Defined
C  {x | x is a positive number}
Not Well Defined
C  {x | x is a positive person}
Well Defined and Not Well Defined
Sets
On your own, come up with one example of a
well defined set and one example of a not well
defined set. Place your sets in the appropriate
section of the board.
Elements of Sets
The symbol  means that the object in question is
an element of a particular set.
The symbol  means that the object in question is
NOT an element of a particular set.
Elements of Sets
For example:
a {w, a, s, h, e, r}
a {m, i, x, e, r}
Since sets can not be ELEMENTS of other sets
{a} {w, a, s, h, e, r}
Homework
• Do page 55 #53 – 84
• QUIZ tomorrow on pages 54-55 #1 - 84
Homework Answers
53. Well Defined
54. Well Defined
55. Not Well Defined
56. Not Well Defined
57. Not Well Defined
58. Well Defined
59. 
60. 
61. 
62. 
63. 
64. 
65. 
66. 
67.False
68.False
69.True
70.True
71.True
72.True
73.True
74.True
75.False
76.False
77.True
78.True
79.True
80.True
81.False
82.False
83.True
84.False
Tuesday Oct 5
• Quiz Today
• After Quiz, do page 56 Question #92.
Finish it for homework and be prepared to
turn it in.
Wednesday Oct 6
Venn Diagrams and Subsets
Consider the set of counting numbers less than or
equal to 20.
U={x|x is a counting number less than 20}
U={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20}
Use the following Venn Diagram to divide the
numbers into groups of even numbers and
groups of multiples of three.
Counting Numbers ≤ 20
Even Numbers
Multiples of 3
U
Counting Numbers ≤ 20
1
Even Numbers
7
Multiples of 3
U
11
5
3
2
8
4
10
6
12
14
16
9
13
15
18
20
19
17
Wednesday Oct 6
Venn Diagrams and Subsets
Universal Set – The set of all objects under
discussion. For our example, the
universal set is the set of all counting
numbers less than or equal to 20. The
universal set is always denoted by the
letter U
Wednesday Oct 6
Venn Diagrams and Subsets
Let’s let A represent the set of all even
numbers less than or equal to 20 and B
will represent the set of all multiples of 3
that are less than or equal to 20
A = {2,4,6,8,10,12,14,16,18,20}
B = {3,6,9,12,15,18}
Counting Numbers ≤ 20
1
Even Numbers
5
11
A
2
10
3
6
8
4
12
14
16
7
Multiples of 3
U
B
9
13
15
18
20
19
17
Wednesday Oct 6
Venn Diagrams and Subsets
Complement of a Set – The complement of
a set is the set of all elements of the
universal set that are NOT elements of
the set in question. In our example the
complement of A, written A´, is
A´={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
Counting Numbers ≤ 20
1
Even Numbers
5
11
A
2
10
3
6
8
4
12
14
16
7
Multiples of 3
U
B
9
13
15
18
20
19
17
Wednesday Oct 6
Venn Diagrams and Subsets
Subset of a Set – The subset of a set is the set
where ALL elements of one set are also
elements another set.
Using our example, A is a subset of U. B is also
a subset of U
Wednesday Oct 6
Venn Diagrams and Subsets
U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
A = {2,4,6,8,10,12,14,16,18,20}
U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
B = {3,6,9,12,15,18}
Wednesday Oct 6
Venn Diagrams and Subsets
A  U means "A is a subset of U"
Is B  U ? Why or why not?
Wednesday Oct 6
Venn Diagrams and Subsets
A  U means "A is a subset of U"
Is B  U ? Why or why not?
Yes, because every element of B
is also an element of U
Wednesday Oct 6
Venn Diagrams and Subsets
Is A  B ? Is B  A ? Why or why not?
Wednesday Oct 6
Venn Diagrams and Subsets
Is A  B ? Is B  A ? Why or why not?
No because NOT EVERY element of A
is an element of B
No because NOT EVERY element of B
is an element of A
Counting Numbers ≤ 20
1
Even Numbers
5
11
A
2
10
3
6
8
4
12
14
16
7
Multiples of 3
U
B
9
13
15
18
20
19
17
Wednesday Oct 6
Venn Diagrams and Subsets
Example: Consider the following
U  {a, b, c, d , e,...x, y, z}
D  {s, a, w, y, e, r}
F  { y, e, s}
Is D  F ?
Is F  D ?
Wednesday Oct 6
Venn Diagrams and Subsets
Example: Consider the following
U  {a, b, c, d , e,...x, y, z}
D  {s, a, w, y, e, r}
F  { y, e, s}
Is D  F ? No, because s, w and d  F
Is F  D ? Yes, because y, e and s  D
Wednesday Oct 6
Homework
Page 61 #1 – 14
Subsets and Proper Subsets
List all of the subsets
{ } or Ø
Subsets and Proper Subsets
List all of the subsets
{ } or Ø
Ø
Subsets and Proper Subsets
List all of the subsets
{a}
Subsets and Proper Subsets
List all of the subsets
{a}
Ø
{a}
Subsets and Proper Subsets
List all of the subsets
{a, b}
Subsets and Proper Subsets
List all of the subsets
{a, b}
Ø
{a}
{b}
{a, b}
Subsets and Proper Subsets
List all of the subsets
{a, b, c}
Subsets and Proper Subsets
List all of the subsets
{a, b, c }
Ø
{a}; {b}; {c}
{a, b}; {a, c}; {b, c}
{a, b, c }
Subsets and Proper Subsets
List all of the subsets
{a, b, c, d}
Subsets and Proper Subsets
List all of the subsets
{a, b, c, d }
Ø
{a}; {b}; {c}; {d}
{a, b}; {a, c}; {a, d}; {b, c}; {b, d}; {c, d}
{a, b, c }; {a, b, d }; {a, c, d }; {b, c, d }
{a, b, c, d }
Subsets and Proper Subsets
List all of the proper subsets
{ } or Ø
Subsets and Proper Subsets
List all of the proper subsets
{ } or Ø
Subsets and Proper Subsets
List all of the proper subsets
{a}
Subsets and Proper Subsets
List all of the proper subsets
{a}
Ø
Subsets and Proper Subsets
List all of the proper subsets
{a, b}
Subsets and Proper Subsets
List all of the proper subsets
{a, b}
Ø
{a}
{b}
Subsets and Proper Subsets
List all of the proper subsets
{a, b, c}
Subsets and Proper Subsets
List all of the proper subsets
{a, b, c }
Ø
{a}; {b}; {c}
{a, b}; {a, c}; {b, c}
Subsets and Proper Subsets
List all of the proper subsets
{a, b, c, d}
Subsets and Proper Subsets
List all of the proper subsets
{a, b, c, d }
Ø
{a}; {b}; {c}; {d}
{a, b}; {a, c}; {a, d}; {b, c}; {b, d}; {c, d}
{a, b, c }; {a, b, d }; {a, c, d }; {b, c, d }
Subsets and Proper Subsets
a set that contains n elements will have
n
2 subsets
and
(2n  1) proper subsets
Homework Page 61-62 #15-53
Homework Page 61-62 #15-53
15. bot h
17. 
19. bot h
21. neit her
23. T rue
25. False
27. T rue
29. T rue
31. T rue
33. T rue
35. False
37. False
39. T rue
41. False
43. 8 subsets
7 propersubsets
45. 64 subsets
63 propersubsets
47. 32 subsets
31 propersubsets
49. {2, 3, 5, 7, 9,10}
51. {2}
53. U  {1, 2, 3, 4, 5, 6, 7, 8, 9,10}
Page 62 #55 - 60
Page 62 #55 - 60
U = {Higher Cost, Lower Cost, Educational, More
Time to See Sights, Less Time to See Sights, Cannot
Visit Relatives, Can Visit Relatives}
Page 62 #55 - 60
U = {Higher Cost, Lower Cost, Educational, More
Time to See Sights, Less Time to See Sights, Cannot
Visit Relatives, Can Visit Relatives}
F = {Higher Cost, Educational, More Time to See
Sights, Cannot Visit Relatives}
F΄ = {Lower Cost, Less Time to See Sights, Can Visit
Relatives}
Page 62 #55 - 60
U = {Higher Cost, Lower Cost, Educational, More
Time to See Sights, Less Time to See Sights, Cannot
Visit Relatives, Can Visit Relatives}
D = {Lower Cost, Educational, Less Time to See
Sights, Can Visit Relatives}
D΄ = {Higher Cost, More Time to See Sights, Cannot
Visit Relatives}
Page 62 #55 - 60
F = {Higher Cost, Educational, More Time to See
Sights, Cannot Visit Relatives}
D = {Lower Cost, Educational, Less Time to See
Sights, Can Visit Relatives}
Both F and D = {Educational}
Page 62 #55 - 60
F΄ = {Lower Cost, Less Time to See Sights, Can Visit
Relatives}
D΄ = {Higher Cost, More Time to See Sights, Cannot
Visit Relatives}
Both F΄ and D ΄ = Ø
Page 62 #55 - 60
F = {Higher Cost, Educational, More Time to See
Sights, Cannot Visit Relatives}
D΄ = {Higher Cost, More Time to See Sights, Cannot
Visit Relatives}
Both F and D ΄ = {Higher Cost, More Time to See
Sights, Cannot Visit Relatives}