STRUCTURE OF ALGEBRA
(Prelimenaries)
Drs. Antonius Cahya Prihandoko, M.App.Sc
Mathematics Education Study Program
Faculty of Teacher Training and Education the University of Jember
Indonesia
Outline
1 Set Theory
Set Subset
2 Equivalence Relation and Partition
Partition
Equivalence Relation
3 Function and Binary Operation
Function
Outline
1 Set Theory
Set Subset
2 Equivalence Relation and Partition
Partition
Equivalence Relation
3 Function and Binary Operation
Function
Outline
1 Set Theory
Set Subset
2 Equivalence Relation and Partition
Partition
Equivalence Relation
3 Function and Binary Operation
Function
About Set
1 A setSconsists of elements, and ifais an element ofS,
then it can be notated asa∈S.
2 There is exactly one set that has no element. It is called empty set, dan be notated asφ.
3 A set can be described by identifying its properties, or by listing its elements. For example, set of prime numbers less than or equal to 5, can be described as{2,3,5}, or {x|x primes≤5}.
4 A set is calledwell-defined, if it can be determined definitely whether an object is element or not. Let
About Set
1 A setSconsists of elements, and ifais an element ofS,
then it can be notated asa∈S.
2 There is exactly one set that has no element. It is called
empty set, dan be notated asφ.
3 A set can be described by identifying its properties, or by listing its elements. For example, set of prime numbers less than or equal to 5, can be described as{2,3,5}, or {x|x primes≤5}.
4 A set is calledwell-defined, if it can be determined definitely whether an object is element or not. Let
About Set
1 A setSconsists of elements, and ifais an element ofS,
then it can be notated asa∈S.
2 There is exactly one set that has no element. It is called
empty set, dan be notated asφ.
3 A set can be described by identifying its properties, or by
listing its elements. For example, set of prime numbers less than or equal to 5, can be described as{2,3,5}, or
{x|x primes≤5}.
4 A set is calledwell-defined, if it can be determined definitely whether an object is element or not. Let
About Set
1 A setSconsists of elements, and ifais an element ofS,
then it can be notated asa∈S.
2 There is exactly one set that has no element. It is called
empty set, dan be notated asφ.
3 A set can be described by identifying its properties, or by
listing its elements. For example, set of prime numbers less than or equal to 5, can be described as{2,3,5}, or
{x|x primes≤5}.
4 A set is calledwell-defined, if it can be determined
definitely whether an object is element or not. Let
S={some natural numbers}, thenSis notwell-defined
set because it can not be decided whether 5∈S or 56∈S.
IfS={the first four natural numbers}, then elements ofS
Subset
Definition
A setB issubsetof setAand be notated ”B ⊆A” or ”A⊇B”, if
every element ofBis also element ofA.
Note
For every setA, Both ofAandφare subset ofA.Aisimproper subset, while the others areproper subset
Example
Subset
Definition
A setB issubsetof setAand be notated ”B ⊆A” or ”A⊇B”, if
every element ofBis also element ofA.
Note
For every setA, Both ofAandφare subset ofA.Aisimproper
subset, while the others areproper subset
Example
Subset
Definition
A setB issubsetof setAand be notated ”B ⊆A” or ”A⊇B”, if
every element ofBis also element ofA.
Note
For every setA, Both ofAandφare subset ofA.Aisimproper
subset, while the others areproper subset
Example
LetS={a,b,c}, thenS has 8 subsets that isφ,{a},{b},{c},
Partition
Definition
Apartitionof setAis a family set consisting of disjoint non
empty subsets which is union of them constructs the setA
How to prove a partition?
Based on that definition, proving that a family set
{A1,A2,A3, ...,An}is a partition of setA, can be shown below :
1 ∀i,j∈ {1,2,3, ...,n}, ifi 6=j thenAi∩Aj =φ;
2 ∪n
Partition
Definition
Apartitionof setAis a family set consisting of disjoint non
empty subsets which is union of them constructs the setA
How to prove a partition?
Based on that definition, proving that a family set
{A1,A2,A3, ...,An}is a partition of setA, can be shown below :
1 ∀i,j∈ {1,2,3, ...,n}, ifi =6 j thenAi∩Aj =φ;
2 ∪n
Partition
Definition
Apartitionof setAis a family set consisting of disjoint non
empty subsets which is union of them constructs the setA
How to prove a partition?
Based on that definition, proving that a family set
{A1,A2,A3, ...,An}is a partition of setA, can be shown below :
1 ∀i,j∈ {1,2,3, ...,n}, ifi 6=j thenAi∩Aj =φ;
2 ∪n
Partition
Definition
Apartitionof setAis a family set consisting of disjoint non
empty subsets which is union of them constructs the setA
How to prove a partition?
Based on that definition, proving that a family set
{A1,A2,A3, ...,An}is a partition of setA, can be shown below :
1 ∀i,j∈ {1,2,3, ...,n}, ifi 6=j thenAi∩Aj =φ;
2 ∪n
Equivalence Relation
element ofA. Set of all elements equivalent tox is calledequivalence class of x, and be denoted by[x]. It is formally written as
[x] ={a∈A|a∼x}
Equivalence Relation
element ofA. Set of all elements equivalent tox is calledequivalence class of x, and be denoted by[x]. It is formally written as
[x] ={a∈A|a∼x}
Equivalence Relation
element ofA. Set of all elements equivalent tox is calledequivalence class of x, and be denoted by[x]. It is formally written as
[x] ={a∈A|a∼x}
Equivalence Relation
element ofA. Set of all elements equivalent tox is calledequivalence class of x, and be denoted by[x]. It is formally written as
[x] ={a∈A|a∼x}
Equivalence Relation
element ofA. Set of all elements equivalent tox is calledequivalence class of x, and be denoted by[x]. It is formally written as
[x] ={a∈A|a∼x}
Equivalence Relation
Theorem 1
Ifx ∼y then[x] = [y].
Theorem 2
If ”∼” is an equivalence relation on a setA, then the set of all equivalence classes,{[x]|x ∈A}, forms a partition onA
Modulo Class
Letaandbbe two integers onZandnbe any positive integer. ais congruent tobmodulon, and be denoteda≡b(modn), if a−bis evenly divisible byn, so thata−b=nk, for some k ∈ Z. Equivalence classes for congruence modulo nare
Equivalence Relation
Theorem 1
Ifx ∼y then[x] = [y].
Theorem 2
If ”∼” is an equivalence relation on a setA, then the set of all
equivalence classes,{[x]|x ∈A}, forms a partition onA
Modulo Class
Letaandbbe two integers onZandnbe any positive integer. ais congruent tobmodulon, and be denoteda≡b(modn), if a−bis evenly divisible byn, so thata−b=nk, for some k ∈ Z. Equivalence classes for congruence modulo nare
Equivalence Relation
Theorem 1
Ifx ∼y then[x] = [y].
Theorem 2
If ”∼” is an equivalence relation on a setA, then the set of all
equivalence classes,{[x]|x ∈A}, forms a partition onA
Modulo Class
Letaandbbe two integers onZandnbe any positive integer.
ais congruent tobmodulon, and be denoteda≡b(modn), if
a−bis evenly divisible byn, so thata−b=nk, for some
k ∈ Z. Equivalence classes for congruence modulo nare
Function
Definition
Afunctionφfrom a setAto a setB is a relation that assigns all elements ofAinto exactly one element ofB.
Type of Function
A function from a setAto a setBisinjectiveif for each element ofB there is at most one element ofAthat be
Function
Definition
Afunctionφfrom a setAto a setB is a relation that assigns all elements ofAinto exactly one element ofB.
Type of Function
A function from a setAto a setBisinjectiveif for each
element ofB there is at most one element ofAthat be
connected to it; and issurjectiveif for each element ofBthere
Binary Operation
Definition
Binary operationon a set,S, is a function that assigns each ordered pair of elements ofS,(a,b)to an element ofS.
Special Binary Operation
A binary operation∗on a setSiscommutativeif and only if a∗b=b∗a,∀a,b∈S; andassociativeif and only if
Binary Operation
Definition
Binary operationon a set,S, is a function that assigns each ordered pair of elements ofS,(a,b)to an element ofS.
Special Binary Operation
A binary operation∗on a setSiscommutativeif and only if
a∗b=b∗a,∀a,b∈S; andassociativeif and only if
