set theory symbols and terminology set – a collection of objects
TRANSCRIPT
Set TheorySymbols and Terminology
Set – A collection of objects
Set TheorySymbols and Terminology
Element – An object in a set
Set TheorySymbols and Terminology
Empty (Null) Set – A set that contains no elements
Set TheorySymbols and Terminology
Cardinal Number (Cardinality) – The number of elements in a set
Set TheorySymbols and Terminology
Finite Set – A set that contains a limited number of elements
Set TheorySymbols and Terminology
Infinite Set – A set that contains an unlimited number of elements
Set TheorySymbols and Terminology
There are three ways to describe a set
Word Description
Listing
Set Builder Notation
Set TheorySymbols and Terminology
The following example shows the three ways we can describe the same set.
Set TheorySymbols and Terminology
Word Description“The set of even counting numbers less
than 10”
Set TheorySymbols and Terminology
ListingE = {2, 4, 6, 8 }
Set TheorySymbols and Terminology
Set Builder NotationE = {x | x is an even counting number
that is less than 10 }
Set TheorySymbols and Terminology
Cardinal Numbersn(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 UpFind the cardinal number of each set
K = {2, 4, 8, 16}
M = { 0 }
R = {4, 5, …, 12, 13}
P = Ø
Warm UpFind 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 TheorySymbols and Terminology
Empty Set – Example)
P = {x | x is a positive number <0}
Therefore P = { } or P = Ø but
P ≠ {Ø}
Set TheorySymbols and Terminology
Infinite Set – Example)
R = {y | y is an odd whole number}
Therefore R = {1, 3, 5, 7, …}
Set TheorySymbols 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 odd15. 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 odd19. 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 odd23. 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 odd31. 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 odd35. 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 odd39. 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
Not We
Well D
ll Defined
{ | is a pretty number}
efined
{ | is a prime number}A x x
A x x
Well Defined and Not Well Defined Sets
Not Well D
Well Defin
efine
ed
{ | is a st
d
{ | is a good student in this cla
udent in this class}
ss}B x x
B x x
Well Defined and Not Well Defined Sets
Not W
Well
ell D
Defined
{
efined
| is a positi
{ | is a positive pe
ve number
son}
}
rC x x
C x x
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 SetsFor example:
Since sets can not be ELEMENTS of
{ , , , ,
{ , , , , }
{
other sets
, }
} { , , , , , }
a w a
a m i x e r
a w a s h
s h e r
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.
68.
69.
70.
71.
False
False
True
True
True
72.
73.
74.
75.
76.
77.
78.
79.
True
True
True
False
False
True
True
True
80.
81.
82.
83.
84.
True
False
False
True
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 6Venn 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.
Even Numbers Multiples of 3
UCounting Numbers Counting Numbers ≤ 20≤ 20
Even Numbers Multiples of 31
2 3
4
6
10
8
12
9
15
5
7
11
13
1719
1416 18
20
UCounting Numbers Counting Numbers ≤ 20≤ 20
Wednesday Oct 6Venn 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 6Venn 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}
Even Numbers Multiples of 31
2 3
4
6
10
8
12
9
15
5
7
11
13
1719
1416 18
20
UCounting Numbers Counting Numbers ≤ 20≤ 20
AB
Wednesday Oct 6Venn 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}
Even Numbers Multiples of 31
3
9
15
5
7
11
13
1719
UCounting Numbers Counting Numbers ≤ 20≤ 20
AB2
4
6
10
8
121416 18
20
Wednesday Oct 6Venn 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 6Venn 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 6Venn Diagrams and Subsets
means "A is a subset of U"
Is ? Why or why not?
A U
B U
Wednesday Oct 6Venn Diagrams and Subsets
Y
means "A is a subset of U"
Is ? Why or why no
es, because every element of B
is also an element of U
t?
A U
B U
Wednesday Oct 6Venn Diagrams and Subsets
Is ? Is ? Why or why not?A B B A
Wednesday Oct 6Venn Diagrams and Subsets
No because NOT EVERY element of A
is an element of B
No because NOT EVER
Is ? Is ?
Y element o
Why o
f B
is
r why
an el
not
ement of A
?A B B A
Even Numbers Multiples of 31
2 3
4
6
10
8
12
9
15
5
7
11
13
1719
1416 18
20
UCounting Numbers Counting Numbers ≤ 20≤ 20
AB
Wednesday Oct 6Venn Diagrams and Subsets
Example: Consider the following
{ , , , , ,... , , }
{ , , , , , }
{ , , }
Is ?
Is ?
U a b c d e x y z
D s a w y e r
F y e s
D F
F D
Wednesday Oct 6Venn Diagrams and SubsetsExample: Consider the following
{ , , , , ,... , , }
{ , , , , , }
{ ,
No, because , and F
Yes, bec
, }
Is ?
Is ? ause , and D
s w d
U a b c d e x y z
D s a w y e r
F y e s
D F
y e sF D
Wednesday Oct 6Homework
Page 61 #1 – 14
Subsets and Proper SubsetsList all of the subsets
{ } or Ø
Subsets and Proper SubsetsList all of the subsets
{ } or Ø
Ø
Subsets and Proper SubsetsList all of the subsets
{a}
Subsets and Proper SubsetsList all of the subsets
{a}
Ø
{a}
Subsets and Proper SubsetsList all of the subsets
{a, b}
Subsets and Proper SubsetsList all of the subsets
{a, b}
Ø
{a}
{b}
{a, b}
Subsets and Proper SubsetsList all of the subsets
{a, b, c}
Subsets and Proper SubsetsList all of the subsets
{a, b, c }
Ø
{a}; {b}; {c}
{a, b}; {a, c}; {b, c}
{a, b, c }
Subsets and Proper SubsetsList all of the subsets
{a, b, c, d}
Subsets and Proper SubsetsList 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 SubsetsList all of the proper subsets
{ } or Ø
Subsets and Proper SubsetsList all of the proper subsets
{ } or Ø
Subsets and Proper SubsetsList all of the proper subsets
{a}
Subsets and Proper SubsetsList all of the proper subsets
{a}
Ø
Subsets and Proper SubsetsList all of the proper subsets
{a, b}
Subsets and Proper SubsetsList all of the proper subsets
{a, b}
Ø
{a}
{b}
Subsets and Proper SubsetsList all of the proper subsets
{a, b, c}
Subsets and Proper SubsetsList all of the proper subsets
{a, b, c }
Ø
{a}; {b}; {c}
{a, b}; {a, c}; {b, c}
Subsets and Proper SubsetsList all of the proper subsets
{a, b, c, d}
Subsets and Proper SubsetsList 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
and
(2 1) prop
2 subsets
er subsets
n
n
Homework Page 61-62 #15-53
Homework Page 61-62 #15-53
False 35.
True 33.
True 31.
True 29.
True 27.
False 25.
True 23.
neither 21.
both 19.
.17
both 15.
10} 9, 8, 7, 6, 5, 4, 3, 2, {1, U53.
{2} 51.
10} 9, 7, 5, 3, {2, 49.
subsetsproper 31
subsets 32 47.
subsetsproper 63
subsets 64 45.
subsetsproper 7
subsets 8 43.
False 41.
True .39
False 37.
Page 62 #55 - 60
Page 62 #55 - 60U = {Higher Cost, Lower Cost, Educational, More Time to See Sights, Less Time to See Sights, Cannot Visit Relatives, Can Visit Relatives}
Page 62 #55 - 60U = {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 - 60U = {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 - 60F = {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 - 60F΄ = {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 - 60F = {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}