virtual field trip: we’ll visit the academic support center, click “my studies” on your ku...
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Virtual Field Trip: We’ll visit the Academic Support Center, click “My Studies” on your KU Campus Homepage.
The pdf and additional materials about Sets can be found at: http://www.ramshillfarm.com/Math/Math150/Unit_2.html
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KU Math Center
Sunday, Wednesday & Thursday: 8:00pm-12:00am (midnight)
Monday: 11:00am-5:00pm AND 8:00pm-12:00am (midnight)
Tuesday: 11:00am-12:00am (midnight)* All times are Eastern Time
Additional Information about the Math Center is in the Doc Sharing Portion of the course.
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2.1
Set Concepts
Page 68
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Set• A set is a collection of objects, which are
called elements or members of the set.
• The symbol , read “is an element of,” is used to indicate membership in a set.
• The symbol , means “is not an element of.”
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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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Roster Form This is the form of the set where the elements
are all listed, each separated by commas.
Description: 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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Set-Builder 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.• |, on the “\” key, is used to denote “such that”.
Description: Set N is the set of all natural numbers less than or equal to 25.
Solution: { x | x є N and 1 ≤ x ≤ 25}
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Cardinal Number
The number of elements in set A is its cardinal number.
Symbol: n(A)A = { 1, 2, 3, 4, 6, 8}n(A) = 6
Page 71
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Empty (or Null) Set
A null set (or empty set ) contains absolutely no elements, and so its cardinal number is 0.
Symbol:
or
Finite Set
A finite set is either empty or the cardinal number is finite.
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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Equivalent Sets
Equivalent sets have the same cardinal number.
Symbol: n(A) = n(B) A = { 1, 3, 5, 7, 9}B = { 2, 4, 6, 8, 10}A & B are equivalent
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Equal sets have the exact same elements in them, regardless of their order.
Symbol: A = B
Example: A = { 1, 2, 4, 5}B = { 2, 5, 4, 1}
Equal Sets
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2.2Subsets
Page 77
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SubsetsA set is a subset of a given set, if everything in the subset is comes from the given set.
Symbol: A BTo 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 .
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.
Note: B is a subset of itself!
Determining Subsets
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Proper Subset
A set is a proper subset of a given set, if it is a subset of that set AND it is smaller than the given set.
Symbol: or
A set can be a Subset, but not a Proper Subset of itself.
Determining Proper Subsets
Example:Determine whether the set A is a proper subset of the 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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Determining Proper Subsets continuedDetermine whether the set A is a proper subset of the 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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Number of Distinct Subsets
The number of distinct subsets of a finite set A is 2n, where n = n(A), the cardinal number of 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 }.
Page 79
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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}, { }
Number of Distinct Subsets continued
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2.3Venn Diagrams
and Set Operations
Page 83
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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.
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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 overlap-
ping area of the two circles.
U
A B
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Intersection
The intersection of two given sets contains only those elements common to both of those sets.
and generally means intersection
Symbol: ABU
A B
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Union
The union of two given sets contains all of the elements for those sets, excluding duplicates.
or generally means union
Symbol: AB
U
A B
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´U
A B
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Subsets
When every element of B is also an element of A.
Circle B is completely inside Circle A.
B A,
28
U
A
B
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The Relationship Between n(A U 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 in the intersection of the sets.
n(A U B) = n(A) + n(B) – n(A ∩ B)
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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
BAU
I II III
IV
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2.4Venn Diagrams with Three Sets
AndVerification of Equality of Sets
Page 95
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General Procedure for Constructing Venn Diagrams with 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}
U A B
C
V
I III
VII
VIIV
VIII
II
32
page 96
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Example: Constructing a Venn diagram for Three Sets completed
33
U A B
C
V
I III
VII
VIIV
VIII
II2
1,85
3
4
6,7
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