Set Notation and DescriptionSet Notation and DescriptionKinds of SetsKinds of Sets

Operations on SetsOperations on Sets

By: Mr. Gerzon B. Mascariñas

GBM 2012

What is set?What is set?

A set is a collection of well-defined objects or things.

Capital letter is used to name a set. An element is an object contained in a

set Example:

A = { 2, 4, 6, 8}

2, 4, 6, and 8 are called elements of set A.

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Set Notation ElementsSet Notation Elements an element is a member of a set notation: means “is an element of”

means “is not an element of” Examples:

– A = {1, 2, 3, 4} 1 A 6 A 2 A z A– B = {x | x is an even number 10}

2 B 9 B 4 B z B

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Set Theory NotationSet Theory NotationSymbol Meaning

Upper case designates set name

Lower case designates set elements

{ } enclose elements in set

or is (or is not) an element of

is a subset of (includes equal sets)

is a proper subset of

is not a subset of

is a superset of

/ or : such that (if a condition is true)

| | the cardinality of a set

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Ways of Describing a SetWays of Describing a Set The roster or listing method:

– A = {2,4,6,8,10}

Descriptive method- Set A is a set of positive even integers less than 12.

Set builder notation method– A = {x|x is a positive even integer less

than 12}– Read as set A is a set of all X such that X

is a positive even integer less than 12

Describe the following:Describe the following:

S = { Math, English, Filipino, Science}

Descriptive method - Set S is a set of major subjects.

Set builder notation method- S = { x/x is a major subject}

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L = {kilogram, hectogram, gram, milligram}

Descriptive method - Set L is a set of metric unit measures

of weight.

Set builder notation method- L = { x/x is a metric unit measures of

weight.}GBM 2012

Identify the elements of the Identify the elements of the following:following:1. Set R is a set of prime numbers

less than 21. - R = { 2, 3, 5, 7, 11, 13, 17, 19 }

2. Set G is a set of factors of 36. - G = { 1, 2, 3, 4, 6, 9, 12 }

3. Set H is a set of vowels of the English alphabet.

- H = { a, e, i, o, u }

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4. Set Y is a set of even numbers greater than 6 but less than 10.

- Y = { 8 }

5. Set M is set of months of the year beginning in M.

- M = { March, May }

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KINDS OF SETSKINDS OF SETS

1. Finite set – a set that has last element or it has a countable number of elements

Example: A = {x/x is a prime number less than

30}The elements of set A can be counted.

There are 10 elements namely 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

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Finite Set CardinalityFinite Set CardinalityCardinality refers to the number of

elements in a set Set Definition Cardinality

A = {x/x is a lower case letter} |A| = 26

B = {2, 3, 4, 5, 6, 7} |B| = 6

C = {x/x is an even number 10} |C|= 4

D = {x/x is an even number 10} |D| = 5

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2. Infinite set – a set with unlimited number of elements or it has at least as many elements as the set of natural numbers.

In listing the elements of an infinite set, we used ellipses (…). This indicates that there are still many elements that follow.

Example:Example:

Set R is a set of even numbers.R = { 2, 4, 6, 8, . . . }

B = {x/x is a whole number}B = {1, 2, 3, 4, 5, 6, 7, . . . }

D = {x/x is a multiple of 10}D = {10, 20, 30, 40, 50, 60, . . .}

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Infinite Set CardinalityInfinite Set Cardinality

Set Definition Cardinality

A = {1, 2, 3, …} |A| =

B = {x | x is a point on a line} |B| =

C = {x| x is a point in a plane} |C| =

0

0

1

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3. Equal set - sets that contain precisely the same elements and same number of elements.

The order in which the elements are listed is unimportant.

Examples:A = {1, 2, 3, 4} B = {1, 4, 2, 3}

A B and B A; therefore, A = B and B = A

A = {c, a, r, e, s} B = {a, r, c, s, e}A B and B A; therefore, A = B and B = A

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4. Equivalent set - sets that have the same number of elements.

Examples:A = {1, 2, 3, 4} B = {p, o , n, d}

Set A has 4 elements, Set B has 4 elements therefore, A B and B A

D = {j, u, n, e} E = {j, u, l, y}Set D has 4 elements, Set E has 4 elements

therefore, A B and B A

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5. Universal set is a set contains all the elements that the other sets have - it is denoted by the symbol U

Example:U = {all students at VPS}

Some Subsets:A = {all Kinder students}B = {Grade school students}C = {Junior high school students}

D = {Senior high school students}

Given:A = {faith, service, love, justice, learning}B = { }C = {justice}D = {love, faith, service}

Set A is a universal set. Set C and D are subsets of A because their

elements can be found in set A.Set B has no element. It is called an empty

set. An empty set is a subset of any set.

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6. Empty set or Null set – a set without an element.

- it is represented by { } or .

7. Subsets - exists when a set’s members are also contained in another set

means “is a subset of”

means “is a proper subset of”

means “is not a subset of”GBM 2012

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Subset RelationshipsSubset Relationships A = {x | x is a positive integer 8}

set A contains: 1, 2, 3, 4, 5, 6, 7, 8 B = {x | x is a positive even integer 10}

set B contains: 2, 4, 6, 8 C = {2, 4, 6, 8, 10}

set C contains: 2, 4, 6, 8, 10 Subset Relationships

A A A B A CB A B B B CC A C B C C

Operations of SetsOperations of Sets1. Union of Sets – denoted by U.

To get the union of the sets, we just put together the elements of the sets without repeating any of the element.

Given: A = {20, 40, 60, 80}B = {10, 30, 50, 70}C = {10, 20, 30, 40}

A U B = {10, 20, 30, 40, 50, 60, 70, 80}A U C = {10, 20, 30, 40, 60, 80}

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2. Intersection of sets – denoted by ∩ - the common element/s between

the given sets

Given: A = {20, 40, 60, 80}B = {10, 30, 50, 70}C = {10, 20, 30, 40}

A ∩ B = { } B ∩ C = {10, 30} A ∩ B ∩ C = { }GBM 2012

3. Complements of Sets3. Complements of Sets Denoted by the symbol (‘) A set of elements which can be

found in the universal set but not in the given set.

Given: U = { 0, 1, 2, . . . 10}A= {0, 1, 2, 3, 4, 5}B= {1, 3, 5, 7, 9}Find:A’ = {6, 7, 8, 9, 10}B’ = {0, 2, 4, 6, 8, 10}(A’)’ = { } GBM 2012

Exercises:Exercises:

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Venn DiagramsVenn Diagrams

Venn diagrams show relationships between sets and their elements

Universal Set

Sets A & B

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Venn Diagram Example 1Venn Diagram Example 1

Set Definition ElementsA = {x | x Z+ and x 8} 1 2 3 4 5 6

7 8B = {x | x Z+; x is even and 10} 2 4

6 8 10A BB A

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Venn Diagram Example 2Venn Diagram Example 2

Set Definition ElementsA = {x | x Z+ and x 9} 1 2 3 4 5

6 7 8 9B = {x | x Z+ ; x is even and 8} 2 4

6 8

A BB AA B

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Venn Diagram Example 3Venn Diagram Example 3

Set Definition ElementsA = {x | x Z+ ; x is even and 10} 2 4

6 8 10B = x Z+ ; x is odd and x 10 } 1 3

5 7 9

A BB A

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Venn Diagram Example 4Venn Diagram Example 4Set Definition

U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 6, 7}B = {2, 3, 4, 7}C = {4, 5, 6, 7}

A = {1, 2, 6, 7}

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Venn Diagram Example 5Venn Diagram Example 5

Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 6, 7}B = {2, 3, 4, 7}C = {4, 5, 6, 7}

B = {2, 3, 4, 7}

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Venn Diagram Example 6Venn Diagram Example 6

Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 6, 7}B = {2, 3, 4, 7}C = {4, 5, 6, 7}

C = {4, 5, 6, 7}

Problem SolvingProblem Solving

200 people entered a carnival. 60 people tried the Octopus, 100 people tried the Ferris Wheel while 40 people tried both the Octopus and the Ferris Wheel.a.How many tried the Octopus only.b.How many tried the Ferris Wheel only?c.How many tried the other rides?

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Set U = 200 peopleLet set O = people tried OctopusLet set F = people tried Ferris wheelLet set R = people tried other rides O F 20 40 60

R 80GBM 2012

The following information was obtained out of 100 grade 7 students.

50 like dancing41 like singing38 like acting10 like dancing, singing, and acting20 like dancing and singing25 like singing and acting15 like dancing and acting

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How many students do not like the three activities?

How many students like dancing only? How many students like singing only? How many students like acting only? How many students like acting and

singing only? How many students like singing and

dancing only? How many students like singing and

acting only?

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A survey result showed that 150 pupils love to read books during their free time and 80 pupils love to play chess. If 70 pupils love both reading and playing chess and 40 pupils do other things, how many pupils participated in the survey?

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