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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 1 Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

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Page 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

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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Page 2: 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

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Page 3: 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

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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Page 4: 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

2.1

Set Concepts

Page 68

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Page 5: 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

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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Page 6: 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

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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Page 7: 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

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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Page 8: 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

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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Page 9: 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

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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Page 10: 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

Empty (or Null) Set

A null set (or empty set ) contains absolutely no elements, and so its cardinal number is 0.

Symbol:

or

Page 11: 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

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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Page 14: 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

2.2Subsets

Page 77

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Page 15: 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

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 .

Page 16: 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

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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Page 17: 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

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.

Page 18: 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

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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Page 19: 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

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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Page 20: 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

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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Page 22: 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

2.3Venn Diagrams

and Set Operations

Page 83

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Page 23: 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

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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Page 24: 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

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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Page 25: 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

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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Page 26: 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

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

Page 27: 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

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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Page 29: 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

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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Page 30: 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

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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Page 31: 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

2.4Venn Diagrams with Three Sets

AndVerification of Equality of Sets

Page 95

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Page 32: 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

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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Page 33: 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

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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