August 2006Copyright © 2006 byDrDelMath.Com 1 Introduction to Sets Basic, Essential, and Important...

August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 11

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

Properties of SetsProperties of Sets

Delano P. Wegener, Ph.D.


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.


Examples - setExamples - set

The collection of persons living in The collection of persons living in Arnold is a set.Arnold is a set. Each person living in Arnold is an Each person living in Arnold is an

element of the set.element of the set.

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

the set.the set.



The collection of all quadrupeds The collection of all quadrupeds is a set.is a set. Each quadruped is an element of the Each quadruped is an element of the

set.set.

The collection of all four-legged The collection of all four-legged dogs is a set.dogs is a set. Each four-legged dog is an element of Each four-legged dog is an element of

the set.the set.



The collection of counting The collection of counting numbers is a set.numbers is a set. Each counting number is an element of Each counting number is an element of

the set.the set.

The collection of pencils in your The collection of pencils in your briefcase is a set.briefcase is a set. Each pencil in your briefcase is an Each pencil in your briefcase is an

element of the set.element of the set.


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.


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.


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}.


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

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


NotationNotation Set builderSet builder notation has the notation has the

general form general form

{variable | descriptive {variable | descriptive statement }.statement }.

The vertical bar (in set builder notation) is The vertical bar (in set builder notation) is always read as “such that”.always read as “such that”.

Set builder notation is frequently used when Set builder notation is frequently used when the roster method is either inappropriate or the roster method is either inappropriate or inadequate.inadequate.


Example – set builder Example – set builder notationnotation

{x | x < 6 and x is a counting {x | x < 6 and x is a counting number} is the set of all counting number} is the set of all counting numbers less than 6. Note this is numbers less than 6. Note this is the same set as {1,2,3,4,5}.the same set as {1,2,3,4,5}.

{x | x is a fraction whose numerator {x | x is a fraction whose numerator is 1 and whose denominator is a is 1 and whose denominator is a counting number }.counting number }.

Set builder notation will become much more concise Set builder notation will become much more concise and precise as more information is introduced.and precise as more information is introduced.


Notation – is an element Notation – is an element ofof

If x is an element of the set A, If x is an element of the set A, we write this as x we write this as x A. x A. x A A means x is not an element of A.means x is not an element of A.

If A = {3, 17, 2 } thenIf A = {3, 17, 2 } then 3 3 A,A, 1717 A, 2 A, 2 A and 5 A and 5 A. A.

If A = { x | x is a prime number } thenIf A = { x | x is a prime number } then 5 5 A, and 6 A, and 6 A. A.


Venn DiagramsVenn Diagrams

It is frequently very helpful to depict aIt is frequently very helpful to depict a

set in the abstract as the points insideset in the abstract as the points inside

a 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


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


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.


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 ..


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.

The set {x | x < 3 and x > 5} is the The set {x | x < 3 and x > 5} is the empty set.empty set.


Definition - subsetDefinition - subset

The set A is a The set A is a subsetsubset of the set of the set B if every element of A is an B if every element of A is an element of B.element of B.

If A is a subset of B and B If A is a subset of B and B contains elements which are not contains elements which are not in A, then A is a in A, then A is a proper subsetproper subset of B.of B.


Notation - subsetNotation - subset

If A is a subset of B we write If A is a subset of B we write A A B to designate that relationship. B to designate that relationship.

If A is a proper subset of B we write If A is a proper subset of B we write A A B to designate that relationship. B to designate that relationship.

If A is not a subset of B we write If A is not a subset of B we write A A B to designate that relationship. B to designate that relationship.


Example - subsetExample - subset

The set A = {1, 2, 3} is a subset of The set A = {1, 2, 3} is a subset of the set B ={1, 2, 3, 4, 5, 6} because the set B ={1, 2, 3, 4, 5, 6} because each element of A is an element of B.each element of A is an element of B.

We write A We write A B to designate this B to designate this relationship between A and B.relationship between A and B.

We could also write We could also write {1, 2, 3} {1, 2, 3} {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6}



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.



The set The set A = {1, 2, 3, 4, 5} is a subset of the set A = {1, 2, 3, 4, 5} is a subset of the set B = {x | x < 6 and x is a counting number}B = {x | x < 6 and x is a counting number}because every element of A is an element because every element of A is an element of B.of B. Notice also that B is a subset of A because every Notice also that B is a subset of A because every element of B is an element of A.element of B is an element of A.


DefinitionDefinition

Two sets A and B are Two sets A and B are equalequal if A if A B and B B and B A. If two sets A and A. If two sets A and B are equal we write A = B to B are equal we write A = B to designate that relationship.designate that relationship.


Example - equalityExample - equalityThe sets The sets

A = {3, 4, 6} and B = {6, 3, 4} A = {3, 4, 6} and B = {6, 3, 4} are are

equal because A equal because A B and B B and B A. A.

The definition of equality of sets shows that The definition of equality of sets shows that the the

order in which elements are written does not order in which elements are written does not

affect the set. affect the set.


Example - equalityExample - equalityIf A = {1, 2, 3, 4, 5} and If A = {1, 2, 3, 4, 5} and

B = {x | x < 6 and x is a counting number} B = {x | x < 6 and x is a counting number}

then then A is a subset of BA is a subset of B because every element because every element

of A is an element of B and of A is an element of B and B is a subset of AB is a subset of A

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

Therefore the two sets are equal and Therefore the two sets are equal and

we write A = B.we write A = B.


Example - equalityExample - equality

The sets A = {2} and B = {2, 5} are not The sets A = {2} and B = {2, 5} are not equal because B is not a subset of A. We equal because B is not a subset of A. We would write A would write A ≠ B. ≠ B. Note that A Note that A B. B.

The sets A = {x | x is a fraction} and The sets A = {x | x is a fraction} and B = {x | x = ¾} are not equal because A B = {x | x = ¾} are not equal because A is not a subset of B. We would write is not a subset of B. We would write A A ≠ B. ≠ B. Note that B Note that B A. A.


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


Example - intersectionExample - intersection

If 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}



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

A A ∩ B = ∩ B = { { , , } }

If A = { If A = { , , , , , , , , , , , , , , , } , }

and B = { and B = { , , , , , , } then } then

A A ∩ B = ∩ B = { { , , , , , , } = B } = B



If A is the set of prime numbers If A is the set of prime numbers and and

B is the set of even numbers then B is the set of even numbers then A A ∩∩ B = { 2 }B = { 2 }

If A = {x | x > 5 } and If A = {x | x > 5 } and B = {x | x < 3 } then B = {x | x < 3 } then A A ∩∩ B = B =



If A = {x | x < 4 } andIf A = {x | x < 4 } and

B = {x | x >1 } thenB = {x | x >1 } then

A A ∩∩ B = {x | 1 < x < 4 }B = {x | 1 < x < 4 }

If A = {x | x > 4 } andIf A = {x | x > 4 } and

B = {x | x >7 } thenB = {x | x >7 } then

A A ∩∩ B = {x | x < 7 }B = {x | x < 7 }


Venn Diagram - Venn Diagram - intersectionintersection

A is represented by the red circle and B A is represented by the red circle and B isis

represented 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 demonstrationdemonstration

http://noppa5.pc.helsinki.fi/koe/boole/boolea.html


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


Example - UnionExample - Union

If 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}.



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

A A B = B = {{, , , , , , , , , , , , , , , , @@, , } }

If A = { If A = { , , , , , , , , } }

and B = {and B = {, , , , } then } then

A A B = B = {{, , , , , , , , } = A } = A



If A is the set of prime numbers and If A is the set of prime numbers and B is the set of even numbers then B is the set of even numbers then A A B = {x | x is even or x is prime }.B = {x | x is even or x is prime }.

If A = {x | x > 5 } and If A = {x | x > 5 } and B = {x | x < 3 } then B = {x | x < 3 } then A A B = {x | x < 3 or x > 5 }. B = {x | x < 3 or x > 5 }.


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

http://noppa5.pc.helsinki.fi/koe/boole/boolea.html


Algebraic PropertiesAlgebraic Properties Union and intersection are Union and intersection are

commutativecommutative operations. operations.

A A B = B B = B A A A A ∩ B = B ∩ A∩ B = B ∩ A

For additional information about the algebra of sets goFor additional information about the algebra of sets go

HEREHERE

http://www.cs.uni.edu/%7Ecampbell/stat/venn.html


Algebraic PropertiesAlgebraic Properties

Union and intersection are Union and intersection are associativeassociative operations. operations.

(A (A B) B) C = A C = A (B (B C) C)

(A (A ∩ B) ∩ C = B ∩ (A ∩ C)∩ B) ∩ C = B ∩ (A ∩ C)


HEREHERE



Two Two distributive distributive laws are true.laws are true.

A A ∩ (∩ ( B B C )= (A C )= (A ∩∩ B) B) (A (A ∩∩ C) C)

A A ( ( B B ∩∩ C )= (A C )= (A B) B) ∩∩ (A (A C) C)


HEREHERE



A few other elementary properties of A few other elementary properties of intersection and union.intersection and union.

A A =A A =A A ∩ ∩ = =

A A A = A A A = A A ∩∩ A = A A = A

For additional information about the algebra of sets For additional information about the algebra of sets gogo HEREHERE


Continued StudyContinued Study

The study of sets is extensive, The study of sets is extensive,

sophisticated, and quite abstract.sophisticated, and quite abstract.

Even at the elementary level manyEven at the elementary level many

considerations have been omitted considerations have been omitted fromfrom

this presentation.this presentation.


Continued StudyContinued Study

For further references check Amazon For further references check Amazon

for books about Set Theory. for books about Set Theory.

Google Set Theory Google Set Theory

The best online resource seems to be this The best online resource seems to be this

Wikipedia page about the Wikipedia page about the Algebra of SetsAlgebra of Sets..

Be sure to follow all the links from that Be sure to follow all the links from that page.page.

Some Elementary Exercises are Some Elementary Exercises are HEREHERE

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

August 2006Copyright © 2006 byDrDelMath.Com 1 Introduction to Sets Basic, Essential, and Important...

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