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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.

Set Concepts

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.”

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.

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.

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}

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

Empty (or Null) Set

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

Symbol:

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.

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

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

2.2Subsets

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

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.

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.

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 }.

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

2.3Venn Diagrams

and Set Operations

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.

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.

Intersection

The intersection of two given sets contains only those elements common to both of those sets.

and generally means intersection

Symbol: ABU

The union of two given sets contains all of the elements for those sets, excluding duplicates.

or generally means union

Symbol: AB

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

Subsets

When every element of B is also an element of A.

Circle B is completely inside Circle A.

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.

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)

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

I II III

2.4Venn Diagrams with Three Sets

AndVerification of Equality of 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}

Example: Constructing a Venn diagram for Three Sets completed

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