August 2006 Copyright © 2006 by DrDelMath.Com 1 Introduction to Sets Basic, Essential, and Important Properties of Sets

August 2006 Copyright Copyright © 2006 by © 2006 by

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Introduction to SetsIntroduction to SetsBasic, Essential, and Important Basic, Essential, and Important

Properties of SetsProperties of Sets


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DefinitionsDefinitions A A setset is a collection of objects. is a collection of objects.

Objects in the collection are Objects in the collection are called called elementselements of the set. of the set.


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Examples - setExamples - set The collection of persons living in The collection of persons living in

Rochester is a set.Rochester is a set. Each person living in Rochester is an Each person living in Rochester is an

element of the set.element of the set.

The collection of all counties in the The collection of all counties in the state of New York is a set.state of New York is a set. Each county in New York is an element of Each county in New York is an element of

the set.the set.


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Notation Notation Sets are usually designated with Sets are usually designated with

capital letterscapital letters..

Elements of a set are usually Elements of a set are usually designated with lower case letters.designated with lower case letters.

We might talk of the set B. An individual We might talk of the set B. An individual element of B might then be designated by b.element of B might then be designated by b.


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NotationNotation The The roster methodroster method of of

specifying a set consists of specifying a set consists of surrounding the collection of surrounding the collection of elements with braces.elements with braces.

Think of a roster as just a comma-Think of a roster as just a comma-separated list enclosed in curly braces.separated list enclosed in curly braces.


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Example – roster methodExample – roster method For example the set of counting For example the set of counting

numbers from 1 to 5 would be numbers from 1 to 5 would be written as written as

{1, 2, 3, 4, 5}. {1, 2, 3, 4, 5}.


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Example – roster methodExample – roster method A variation of the simple roster method A variation of the simple roster method

uses the uses the ellipsis ( … ) when the pattern ( … ) when the pattern is obvious and the set is large.is obvious and the set is large.

{1, 3, 5, 7, … , 9007} is the set of odd {1, 3, 5, 7, … , 9007} is the set of odd counting numbers less than or equal to counting numbers less than or equal to 9007.9007.{1, 2, 3, … } is the set of all counting {1, 2, 3, … } is the set of all counting numbers.numbers.

http://en.wikipedia.org/wiki/Ellipsis


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Venn DiagramsVenn DiagramsIt is frequently very helpful to depict aIt is frequently very helpful to depict aset in the abstract as the points insideset in the abstract as the points insidea circle ( or any other closed shape ).a circle ( or any other closed shape ).

We can picture the set A as We can picture the set A as the points inside the circle the points inside the circle shown here.shown here. A


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Venn DiagramsVenn DiagramsTo learn a bit more about VennTo learn a bit more about Venndiagrams and the man John Venndiagrams and the man John Vennwho first presented these diagramswho first presented these diagramsclick on the history icon at the right.click on the history icon at the right.

History

http://www.andrews.edu/~calkins/math/biograph/biovenn.htm


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Venn DiagramsVenn DiagramsVenn Diagrams are used in Venn Diagrams are used in

mathematics, mathematics, logic, theological ethics, genetics, logic, theological ethics, genetics,

studystudyof Hamlet, linguistics, reasoning, of Hamlet, linguistics, reasoning,

and and many other areas.many other areas.


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DefinitionDefinition The set with no elements is The set with no elements is

called the called the empty setempty set or the null or the null set and is designated with the set and is designated with the symbol symbol ..


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Examples – empty setExamples – empty set The set of all pencils in your The set of all pencils in your

briefcase might indeed be the briefcase might indeed be the empty set.empty set.

The set of even prime numbers The set of even prime numbers greater than 2 is the empty set.greater than 2 is the empty set.


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Definition - subsetDefinition - subset The set A is a The set A is a subsetsubset of the set B if of the set B if

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

Example:Example:

B = {oranges, grapes, apples, mangoes}B = {oranges, grapes, apples, mangoes}A = {grapes, apples}A = {grapes, apples}

A is a subset of BA is a subset of B


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Example - subsetExample - subset The set A = {3, 5, 7} is not a The set A = {3, 5, 7} is not a

subset of the set B = {1, 4, 5, 7, subset of the set B = {1, 4, 5, 7, 9} because 3 is an element of A 9} because 3 is an element of A but is not an element of B.but is not an element of B.

The empty set is a subset of The empty set is a subset of every set, because every every set, because every element of the empty set is an element of the empty set is an element of every other set.element of every other set.


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Definition - intersectionDefinition - intersection The The intersectionintersection of two sets A of two sets A

and B is the set containing those and B is the set containing those elements which are elements which are

elements of A elements of A andand elements of elements of B.B.

We write A We write A B B


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Example - intersectionExample - intersectionIf A = {3, 4, 6, 8} andIf A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then B = { 1, 2, 3, 5, 6} then A A B = {3, 6} B = {3, 6}


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Example - intersectionExample - intersectionIf A = { If A = { , , , , , , , , , , , , , , , , } }and B = { and B = { , , , , , , @@, , , , } then } thenA A ∩ B = ∩ B = { { , , } }

If A = { If A = { , , , , , , , , , , , , , , , } , } and B = { and B = { , , , , , , } then } thenA A ∩ B = ∩ B = { { , , , , , , } = B } = B


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Venn Diagram - Venn Diagram - intersectionintersection

A is represented by the red circle and B isA is represented by the red circle and B isrepresented by the blue circle.represented by the blue circle.When B is moved to overlap a When B is moved to overlap a portion of A, the purple portion of A, the purple colored regioncolored regionillustrates the intersection illustrates the intersection A A ∩ ∩ BB of A and Bof A and BExcellent online interactiveExcellent online interactive demonstration demonstration


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Definition - unionDefinition - union The The unionunion of two sets A and B is of two sets A and B is

the set containing those elements the set containing those elements which are which are

elements of A elements of A oror elements of B. elements of B.

We write A We write A B B


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Example - UnionExample - UnionIf A = {3, 4, 6} andIf A = {3, 4, 6} and B = { 1, 2, 3, 5, 6} then B = { 1, 2, 3, 5, 6} then A A B = {1, 2, 3, 4, 5, 6}. B = {1, 2, 3, 4, 5, 6}.


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Example - UnionExample - UnionIf A = { If A = { , , , , , , , , , , } }and B = { and B = { , , , , , , @@, , , , } then } thenA A B = B = {{, , , , , , , , , , , , , , , , @@, , } }

If A = { If A = { , , , , , , , , } } and B = {and B = {, , , , } then } thenA A B = B = {{, , , , , , , , } = A } = A


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Venn Diagram - unionVenn Diagram - unionA is represented by the red circle and B A is represented by the red circle and B

isisrepresented by the blue circle.represented by the blue circle.The purple colored regionThe purple colored regionillustrates the intersection.illustrates the intersection.The union consists of allThe union consists of allpoints which are colored points which are colored red red oror blue blue oror purple. purple.

Excellent online interactiveExcellent online interactive demonstrationdemonstration

A B

A∩B

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August 2006 Copyright © 2006 by DrDelMath.Com 1 Introduction to Sets Basic, Essential, and Important Properties of Sets