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Chapter 2

Sets

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2.1

Set Concepts

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Set

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

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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.

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.

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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: Reads : Set B is the set of all elements x such that x is a natural number and x is an even number less or equal to 10.

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

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Finite Set

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.

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Infinite 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.

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

Symbol: A = B

Equal Sets

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Cardinal Number

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

Symbol: n(A)

Example Given A = {1,3,5,7,10} find n(A)

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Cardinal Number

n(A) represent the cardinal number of set A, which is the number of elements in set A. Count the number of elements in the set. There are 5 elements in set A.

Therefore, n(A) = 5

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Equivalent Sets

Equivalent sets have the same number of elements in them.

Symbol: n(A) = n(B)

Example:

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Equivalent Sets

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Empty (or Null) Set

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

Symbol: or

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Universal Set

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

Symbol: U

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2.2

Subsets

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Subsets

A set is a subset of a given set if and only if all elements of the subset are also elements of the given set.

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.

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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

Determining Subsets

A B .

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Proper Subset

All subsets are proper subsets except the subset containing all of the given elements, that is, the given set must contain one element not in the subset (the two sets cannot be equal).

Symbol:

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Determining Proper Subsets

Example: Determine whether set A is a proper subset of

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 continued

Example: Determine whether set A is a proper subset of

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 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 }.

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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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2.3

Venn Diagrams and Set Operations

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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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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

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Venn Diagrams: Example

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Venn Diagrams: Example

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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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Intersection

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

Symbol: AB

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Union

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: AB

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Example 1: Determine if a collection is a set.

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Example 1: Determine if a collection is a set.

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Example 2:Determine whether sets are finite or infinite.

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Example 2:Determine whether sets are finite or infinite.

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Example 2:Determine whether sets are finite or infinite.

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Example 3: Determine whether sets are finite or infinite.

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Example 4: Express sets in roster form.

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Example 5: Write descriptions of sets.

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Example 6: Describe complements, unions and intersections based on Venn diagrams.

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SAMPLE 7: Determine elements in intersections, unions and complements.

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