Equivalence relation of discrete--4.4 and partial order relation

4.4 Equivalence relation and partial order relation
• 4.4.1Equivalence relation
• 4.4.2Equivalence classes and quotient sets
• 4.4.3Division of a set
• 4.4.4Partial order Relationship

• 4.4.5 partial sequence set and Hasse

The definition and example of equivalence relation
Defined4.18SetRA relationship on a non-empty collection. IfRis reflexive,
The called and the passing, is saidRForAThe equivalence relation on the. SetRis an equivalent
Relationship, If<x,y>∈R, SaidxEquivalent toy, Remember to dox~y.
Cases1SetA={1, 2, ..., 8},As defined belowAOn the relationshipR：
R={<x,y>|x,y∈A∧x≡y(mod 3)}
whichx≡y(mod 3)CalledxAndyMode3Equal, ThatxDividing3The remainder of the
yDividing3The remainder is equal.
Not difficult to verifyRForAThe equivalence relation on the, Because
x∈A, Yesx≡x(mod 3)
x,y∈A, Ifx≡y(mod 3),Then there arey≡x(mod 3)
x,y,Z∈A, Ifx≡y(mod 3),y≡Z(mod 3),Then there are
x≡Z(mod 3)
Equivalence class
Defined4.19SetRis a non-empty collectionAThe equivalence relation on the, x∈AMake
[x]R= { y| y∈A∧XRy}
Said[x]RForxAboutRThe equivalence class, Jane calledxThe equivalence class, Jiangwei[x].
Instancea={1, 2, ..., 8}Upper Die3Equivalence classes for equivalence relationships:
[1]=[4]=[7]={1,4,7}
[2]=[5]=[8]={2,5,8}
[3]=[6]={3,6}

Business Set
Define 4.20 to set R as the equivalence relation on a non-empty set a , with all R
The equivalence class as a collection of elements is known as a quotient set of R , and a/R is recorded.
A / R = {[x]R | x∈A }

Partial order Relationship
Defines the relationship of reflexive, opposing, and transitive on a non-empty set of 4.22 ,
A partial order relationship called a , which is recorded as ≼. Set ≼ as a partial-order relationship , if
< x , y >∈≼, then x≼y, read x" less than or equal to" y .
Instance
The identity relationship on set a is a partial-order relationship on a .
Less than or equal to the relationship, and the divisible and included relationships are the corresponding collections
On the partial order relationship .

Related concepts
Defined4.23xAndyComparable setRis a non-empty collectionAOn the partial order relationship,
x, yA, xAndyComparable x≼y∨y≼x.
Conclusion: x, yA, one of the following occurs and only one occurs.
x≺y, y≺x(Quasi-sequential), x=y, xAndyIt's not comparable.
Defined4.25FormRis a non-empty collectionAThe partial order on the, x, yA, xAndy
are comparable, it is saidRFor full-order.
Defined4.26Coveredx,y∈A, Ifx≺yAnd does not existZAMakes
x≺Z≺y, is saidyCoveredx.
Instance: a less than or equal relationship on a set of numbers is a full-order relationship
An integer relationship is not a full-order relationship on a set of positive integers
{1, 2, 4, 6}The divisible relationship on the collection, 2Covered1, 4And6Covered2.
But4Do not overwrite1.