Unit 2

Sets

2.1

Set Concepts

A collection of objects, which are called elements or members of the set.

Listing the elements of a set inside a pair of braces, { }, is called roster form.

The symbol є, read “is an element of,” is used to indicate membership in a set.

The symbol є with a line through it, means “is not an element of.”

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.

Well Defined Set

This is the form of the set where the elements are all listed, each separated by commas.

Example:

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.

Roster Form

Set-Builder (or Set-Generator) 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.Example: Write set B = {2, 4, 6, 8, 10} in set-

builder notation.

Solution:

B x x N and x is an even number 10 .

A set that contains no elements or the number of elements in the set is a natural number.

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.

Finite Set

An infinite set contains an indefinite (uncountable) number of elements.

The set of natural numbers is an example of an infinite set because it continues to increase forever without stopping, making it impossible to count its members.

Infinite Set

Equal sets have the exact same elements in them, regardless of their order.

Symbol: A = B

Equal Sets

Cardinal Number

The number of elements in set A is its cardinal number.

Symbol: n(A)

Equivalent sets have the same number of elements in them.

Symbol: n(A) = n(B)

Equivalent Sets

A null set (or empty set ) contains absolutely NO elements.

Symbol:

Empty (or Null) Set

or

The universal set contains all of the possible elements which could be discussed in a particular problem.

Symbol: U

Universal Set

2.2

Subsets

Subsets

Determining Subsets

Proper Subset

Determining Proper Subsets

Determining Proper Subsets Continued

Number of Distinct Subsets

The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set 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 }.

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}, { }

2.3

Venn Diagrams and Set

Operations

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.

Venn Diagrams

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 overlapping area of the two circles.

U

A B

Disjoint Sets

Overlapping Sets

For sets A and B drawn in this figure, notice the overlapping area shared by the two circles.

This section represents the elements that are in the intersection of set A and set B.

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´

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

Symbol:

Intersection

AB

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

The union “unites” that is, it brings together everything into one set.

Symbol:

Union

AB

U

A

B

Subsets

When set A is equal to set B, all the elements of A are elements of B, and all the elements of B are elements of A. Both sets are drawn as one circle.

Equal Sets

U

A B

and is generally interpreted to mean intersection

or is generally interpreted to mean union

The Meaning of and and or

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 common to both sets.

The Relationship Between n(A U B), n(A), n(B), n ( )

n (A U B) = n (A) + n (B) – n ( )

AB

AB

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.

Difference of Two Sets

A B x | x A and x B

A B

U

IV

I II III

The Cartesian product of set A and set B, symbolized A B, and read “A cross B,” is the set of all possible ordered pairs of the form (a, b), where a E A and b E B.

Select the first element of set A and form an ordered pair with each element of set B. Then select the second element of set A and form an ordered pair with each element of set B. Continue in this manner until you have used each element in set A.

Cartesian Product