MAthematics 7 356665408-Introduction-to-Sets.ppt
SalgieSernal2
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About This Presentation
Sets
Slide Content
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 11
Introduction to SetsIntroduction to Sets
Basic, Essential, and Important Basic, Essential, and Important
Properties of SetsProperties of Sets
Prepared by
SALGIE P. SERNAL
Introduction to SetsIntroduction to Sets
Basic, Essential, and Important Basic, Essential, and Important
Properties of SetsProperties of Sets
Prepared by
SALGIE P. SERNAL
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 22
DefinitionsDefinitions
A A setset is a collection of objects. is a collection of objects.
Objects in the collection are called Objects in the collection are called
elementselements of the set. of the set.
DefinitionsDefinitions
A A setset is a collection of objects. is a collection of objects.
Objects in the collection are called Objects in the collection are called
elementselements of the set. of the set.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 33
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 element of Each person living in Arnold is an element of
the set.the set.
The collection of all counties in the The collection of all counties in the
state of Texas is a set.state of Texas is a set.
Each county in Texas is an element of the set.Each county in Texas is an element 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 element of Each person living in Arnold is an element of
the set.the set.
The collection of all counties in the The collection of all counties in the
state of Texas is a set.state of Texas is a set.
Each county in Texas is an element of the set.Each county in Texas is an element of the set.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 44
Examples - setExamples - set
The collection of all quadrupeds is a The collection of all quadrupeds is a
set.set.
Each quadruped is an element of the set.Each quadruped is an element of the 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 the Each four-legged dog is an element of the
set.set.
Examples - setExamples - set
The collection of all quadrupeds is a The collection of all quadrupeds is a
set.set.
Each quadruped is an element of the set.Each quadruped is an element of the 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 the Each four-legged dog is an element of the
set.set.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 55
Examples - setExamples - set
The collection of counting numbers The collection of counting numbers
is a set.is a set.
Each counting number is an element of the Each counting number is an element of the
set.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 element Each pencil in your briefcase is an element
of the set.of the set.
Examples - setExamples - set
The collection of counting numbers The collection of counting numbers
is a set.is a set.
Each counting number is an element of the Each counting number is an element of the
set.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 element Each pencil in your briefcase is an element
of the set.of the set.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 66
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.
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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 77
NotationNotation
The The roster methodroster method of specifying a of specifying a
set consists of surrounding the set consists of surrounding the
collection of elements with braces.collection of elements with braces.
NotationNotation
The The roster methodroster method of specifying a of specifying a
set consists of surrounding the set consists of surrounding the
collection of elements with braces.collection of elements with braces.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 88
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
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}.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 99
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.
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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1010
NotationNotation
Set builderSet builder notation has the notation has the
general form general form
{variable | descriptive statement }.{variable | descriptive 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.
NotationNotation
Set builderSet builder notation has the notation has the
general form general form
{variable | descriptive statement }.{variable | descriptive 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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1111
Example – set builder notationExample – set builder notation
{x | x < 6 and x is a counting number} {x | x < 6 and x is a counting number}
is the set of all counting numbers is the set of all counting numbers
less than 6. Note this is the same set less than 6. Note this is the same set
as {1,2,3,4,5}.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.
Example – set builder notationExample – set builder notation
{x | x < 6 and x is a counting number} {x | x < 6 and x is a counting number}
is the set of all counting numbers is the set of all counting numbers
less than 6. Note this is the same set less than 6. Note this is the same set
as {1,2,3,4,5}.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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1212
Notation – is an element ofNotation – is an element of
If x is an element of the set A, we If x is an element of the set A, we
write this as x write this as x A. x A. x A means x A means x
is not an element of A.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.
Notation – is an element ofNotation – is an element of
If x is an element of the set A, we If x is an element of the set A, we
write this as x write this as x A. x A. x A means x A means x
is not an element of A.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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1313
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 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
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1414
Venn DiagramsVenn Diagrams
To learn a bit more about VennTo learn a bit more about Venn
diagrams and the man John Venndiagrams and the man John Venn
who first presented these diagramswho first presented these diagrams
click on the history icon at the right.click on the history icon at the right.
History
Venn DiagramsVenn Diagrams
To learn a bit more about VennTo learn a bit more about Venn
diagrams and the man John Venndiagrams and the man John Venn
who first presented these diagramswho first presented these diagrams
click on the history icon at the right.click on the history icon at the right.
History
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1515
Venn DiagramsVenn Diagrams
Venn Diagrams are used in Venn Diagrams are used in
mathematics, mathematics,
logic, logic, theological ethics, genetics, , genetics,
studystudy
of Hamlet, linguistics, reasoning, and of Hamlet, linguistics, reasoning, and
many other areas.many other areas.
Venn DiagramsVenn Diagrams
Venn Diagrams are used in Venn Diagrams are used in
mathematics, mathematics,
logic, logic, theological ethics, genetics, , genetics,
studystudy
of Hamlet, linguistics, reasoning, and of Hamlet, linguistics, reasoning, and
many other areas.many other areas.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1616
DefinitionDefinition
The set with no elements is called The set with no elements is called
the the empty setempty set or the null set and is or the null set and is
designated with the symbol designated with the symbol ..
DefinitionDefinition
The set with no elements is called The set with no elements is called
the the empty setempty set or the null set and is or the null set and is
designated with the symbol designated with the symbol ..
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1717
Examples – empty setExamples – empty set
The set of all pencils in your briefcase The set of all pencils in your briefcase
might indeed be the empty set.might indeed be the 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.
Examples – empty setExamples – empty set
The set of all pencils in your briefcase The set of all pencils in your briefcase
might indeed be the empty set.might indeed be the 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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1818
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.
If A is a subset of B and B contains If A is a subset of B and B contains
elements which are not in A, then A elements which are not in A, then A
is a is a proper subsetproper subset of B. of B.
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.
If A is a subset of B and B contains If A is a subset of B and B contains
elements which are not in A, then A elements which are not in A, then A
is a is a proper subsetproper subset of B. of B.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 1919
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.
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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2020
Example - subsetExample - subset
The set A = {1, 2, 3} is a subset of the set The set A = {1, 2, 3} is a subset of the set
B ={1, 2, 3, 4, 5, 6} because each B ={1, 2, 3, 4, 5, 6} because each
element of A is an element of B.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}
Example - subsetExample - subset
The set A = {1, 2, 3} is a subset of the set The set A = {1, 2, 3} is a subset of the set
B ={1, 2, 3, 4, 5, 6} because each B ={1, 2, 3, 4, 5, 6} because each
element of A is an element of B.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}
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2121
Example - subsetExample - subset
The set A = {3, 5, 7} is not a subset The set A = {3, 5, 7} is not a subset
of the set B = {1, 4, 5, 7, 9} because of the set B = {1, 4, 5, 7, 9} because
3 is an element of A but is not an 3 is an element of A but is not an
element of B.element of B.
The empty set is a subset of every The empty set is a subset of every
set, because every element of the set, because every element of the
empty set is an element of every empty set is an element of every
other set.other set.
Example - subsetExample - subset
The set A = {3, 5, 7} is not a subset The set A = {3, 5, 7} is not a subset
of the set B = {1, 4, 5, 7, 9} because of the set B = {1, 4, 5, 7, 9} because
3 is an element of A but is not an 3 is an element of A but is not an
element of B.element of B.
The empty set is a subset of every The empty set is a subset of every
set, because every element of the set, because every element of the
empty set is an element of every empty set is an element of every
other set.other set.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2222
Example - subsetExample - subset
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.
Example - subsetExample - subset
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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2323
DefinitionDefinition
Two sets A and B are Two sets A and B are equalequal if A if A B B
and B and B A. If two sets A and B are A. If two sets A and B are
equal we write A = B to designate equal we write A = B to designate
that relationship.that relationship.
DefinitionDefinition
Two sets A and B are Two sets A and B are equalequal if A if A B B
and B and B A. If two sets A and B are A. If two sets A and B are
equal we write A = B to designate equal we write A = B to designate
that relationship.that relationship.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2424
Example - equalityExample - equality
The sets The sets
A = {3, 4, 6} and B = {6, 3, 4} are A = {3, 4, 6} and B = {6, 3, 4} are
equal because A equal because A B and B B and B A. A.
The definition of equality of sets shows that the The definition of equality of sets shows that 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 - equality
The sets The sets
A = {3, 4, 6} and B = {6, 3, 4} are A = {3, 4, 6} and B = {6, 3, 4} are
equal because A equal because A B and B B and B A. A.
The definition of equality of sets shows that the The definition of equality of sets shows that the
order in which elements are written does not order in which elements are written does not
affect the set. affect the set.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2525
Example - equalityExample - equality
If 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
If 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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2626
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.
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.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2727
Definition - intersectionDefinition - intersection
The The intersectionintersection of two sets A and of two sets A and
B is the set containing those B is the set containing those
elements which are elements which are
elements of A elements of A andand elements of B. elements of B.
We write A We write A B B
Definition - intersectionDefinition - intersection
The The intersectionintersection of two sets A and of two sets A and
B is the set containing those B is the set containing those
elements which are elements which are
elements of A elements of A andand elements of B. elements of B.
We write A We write A B B
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2828
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}
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}
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 2929
Example - intersectionExample - intersection
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
Example - intersectionExample - intersection
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
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3030
Example - intersectionExample - intersection
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 = { 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 =
Example - intersectionExample - intersection
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 = { 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 =
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3131
Example - intersectionExample - intersection
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 }
Example - intersectionExample - intersection
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 }
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3232
Venn Diagram - intersectionVenn Diagram - intersection
A is represented by the red circle and B isA is represented by the red circle and B is
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 region
illustrates the intersection illustrates the intersection
A A ∩ ∩ BB
of A and Bof A and B
Excellent online interactiveExcellent online interactive demonstration demonstration
Venn Diagram - intersectionVenn Diagram - intersection
A is represented by the red circle and B isA is represented by the red circle and B is
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 region
illustrates the intersection illustrates the intersection
A A ∩ ∩ BB
of A and Bof A and B
Excellent online interactiveExcellent online interactive demonstration demonstration
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3333
Definition - unionDefinition - union
The The unionunion of two sets A and B is the of two sets A and B is the
set containing those elements which set containing those elements which
are are
elements of A elements of A oror elements of B. elements of B.
We write A We write A B B
Definition - unionDefinition - union
The The unionunion of two sets A and B is the of two sets A and B is the
set containing those elements which set containing those elements which
are are
elements of A elements of A oror elements of B. elements of B.
We write A We write A B B
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3434
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}.
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}.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3535
Example - UnionExample - Union
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
Example - UnionExample - Union
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
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3636
Example - UnionExample - Union
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 }.
Example - UnionExample - Union
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 }.
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3737
Venn Diagram - unionVenn Diagram - union
A is represented by the red circle and B isA is represented by the red circle and B is
represented by the blue circle.represented by the blue circle.
The purple colored regionThe purple colored region
illustrates the intersection.illustrates the intersection.
The union consists of allThe union consists of all
points 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
Venn Diagram - unionVenn Diagram - union
A is represented by the red circle and B isA is represented by the red circle and B is
represented by the blue circle.represented by the blue circle.
The purple colored regionThe purple colored region
illustrates the intersection.illustrates the intersection.
The union consists of allThe union consists of all
points 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
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3838
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
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
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 3939
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)
∩ ∩ ∩ ∩
For additional information about the algebra of sets goFor additional information about the algebra of sets go HEREHERE
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)
∩ ∩ ∩ ∩
For additional information about the algebra of sets goFor additional information about the algebra of sets go HEREHERE
August 2006August 2006 Copyright Copyright © 2006 by © 2006 by DrDelMath.Com DrDelMath.Com 4040
Algebraic PropertiesAlgebraic Properties
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)
For additional information about the algebra of sets goFor additional information about the algebra of sets go
HEREHERE
Algebraic PropertiesAlgebraic Properties
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)
For additional information about the algebra of sets goFor additional information about the algebra of sets go
HEREHERE
Algebraic PropertiesAlgebraic Properties
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 goFor additional information about the algebra of sets go
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 goFor additional information about the algebra of sets go
HEREHERE
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