1. 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:
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2. Can't Type. press F11 or F5 Can’t Hear
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3. KU Math Center
Sunday, Wednesday & Thursday: 8:00pm-12:00am (midnight) Monday: 11:00am-5:00pm AND 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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4. Page 68
2.1
Set Concepts
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5. 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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6. 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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7. 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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8. 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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9. Cardinal Number
Page 71
The number of elements in set A is its cardinal number.
Symbol: n(A)
A = { 1, 2, 3, 4, 6, 8}
n(A) = 6
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10. Empty (or Null) Set
A null set (or empty set ) contains absolutely no elements, and so its cardinal number is 0.
Symbol:

11. 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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12. 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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13. Equal Sets
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}
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14. Page 77
2.2
Subsets
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15. Subsets
A set is a subset of a given set, if everything in the subset is comes from the given set.
Symbol: A 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 .

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.
Note: B is a subset of itself! Í
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17. 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.

18. 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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19. Determining Proper Subsets continued
Determine 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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20. Number of Distinct Subsets
Page 79
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 }.
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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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22. 2.3 Venn Diagrams and Set Operations Page 83
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23. 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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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
are no elements in common
since there is no overlap-
ping area of the two circles.
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25. Intersection
The intersection of two given sets contains only those elements common to both of those sets.
and generally means intersection
Symbol:
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26. Union
The union of two given sets contains all of the elements for those sets, excluding duplicates.
or generally means union
Symbol:

27. 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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28. Subsets When every element of B is also an element of A.
Circle B is completely inside Circle A. U A B
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29. 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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30. 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. B A U I II
III IV
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31. 2.4 Venn Diagrams with Three Sets And Verification of Equality of Sets
Page 95
2.4
Venn Diagrams with Three Sets
And
Verification of Equality of Sets
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32. General Procedure for Constructing Venn Diagrams with Three Sets
page 96
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 VI IV
VIII II
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33. Example: Constructing a Venn diagram for Three Sets completedB C V I
III
VII VI IV
VIII II 4 2 5
1,8 3
6,7
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