2.6 Equivalence Relation

1.Equivalence relationDefinition 2.18: A relation R on a set A is

called an equivalence relation if it is reflexive, symmetric, and transitive.

Example: Let m be a positive integer with m>1. Show that congruence modulo m is an equivalence relation. R={(a,b)|ab mod m}

Proof: (1)reflexive (for any aZ ， aRa?)(2)symmetric (for any aRb ， bRa?)(3)transitive (for aRb ， bRc ， aRc?)

2.Equivalence classespartitionDefinition 2.19: A partition or quotient set of a

nonempty set A is a collection of nonempty subsets of A such that

(1)Each element of A belongs to one of the sets in . (2)If Ai and Aj are distinct elements of , then

Ai∩Aj=.The sets in are called the bocks or cells of the

partition.Example: Let A={a,b,c},P={{a,b},{c}},S={{a},{b},{c}},T={{a,b,c}},U={{a},{c}},V={{a,b},{b,c}},W={{a,b},{a,c},{c}}, infinite

Example ： congruence modulo 2 is an equivalence relation.

For any xZ, or x=0 mod 2,or x=1 mod 2, i.e or xE ,or xO.

And E∩O=E and O ，{E, O} is a partition of Z

Definition 2.20: Let R be an equivalence relation on a set A. The set of all element that are related to an element a of A is called the equivalence class of a. The equivalence class of a with respect to R is denoted by [a]R, When only one relation is under consideration, we will delete the subscript R and write [a] for this equivalence class.

Example ： equivalence classes of congruence modulo 2 are [0] and [1] 。

[0]={…,-4,-2,0,2,4,…}=[2]=[4]=[-2]=[-4]=…[1]={…,-3,-1,1,3,…}=[3]=[-1]=[-3]=…the partition of Z =Z/R={[0],[1]}

Example: equivalence classes of congruence modulo n are ：

[0]={…,-2n,-n,0,n,2n,…}[1]={…,-2n+1,-n+1,1,n+1,2n+1,…}…[n-1]={…,-n-1,-1,n-1,2n-1,3n-1,…}A partition or quotient set of Z,Z/R={[0],[1],…,[n-1]}

Theorem 2.11 ： Let R be an equivalence relation on A. Then

(1)For any aA, a[a]; (2)If a R b, then [a]=[b]; (3)For a,bA, If (a,b)R, then [a]∩[b]=;

Proof ： (1)For any aA ， aRa? (2)For a,bA, aRb, [a]?[b] ， [b]?[a]For any x[a] ,x?[b] when aRb ， i.e. x R b for any x[b], x?[a] when aRb ， i,.e.xRa (3)For a,bA, If (a,b)R, then [a]∩[b]=Reduction to absurditySuppose [a]∩[b]≠, Then there exists x[a]∩[b]. (4)

AaAa

][)4(

The equivalence classes of an equivalence relation on a set form a partition of the set.

Equivalence relation partitionExample ： Let A={1,2,3,4}, and let

R={(1,1),(2,2),(3,3),(4,4), (1,3),(2,4),(3,1),(4,2)} is an equivalence relation.

Then the equivalence classes are:

Conversely, every partition on a set can be used to form an equivalence relation.

Let ={A1,A2,…,An} be a partition of a nonempty set A. Let R be a relation on A, and aRb if only if there exists Ai s.t. a,bAi.

i.e. R=(A1A1) (A∪ 2A2) … (A∪ ∪ nAn)R is an equivalence relation on ATheorem 2.12 ： Given a partition {Ai|iZ}

of the set A, there is an equivalence relation R that has the set Ai, iZ, as its equivalence classes

Example: Let ={{a,b},{c}} be a partition of A={a,b,c}.

Equivalence relation R=?

2.7 Partial order relations

1.Partially ordered setsDefinition 2.21: A relation R on a set A is called a

partial order if R is reflexive, antisymmetric, and transitive. The set A together with the partial order R is called a partially ordered set, or simply a poset, and we will denote this poset by (A,R). And the notation a b denoteds that (a,b)≼ R. Note that the symbol is used to denote the relation in any ≼poset, not just the “lessthan or equals” relation. The notation a b denotes that a b but a≺ ≼ b.

The relation on R≦ ；The relation | on Z+ ； the relation on P(A) 。partial order ， (R, ), (Z≦ +,/), (P(A),) are partially ordered sets 。Example ： Let A={1,2},P(A)={,{1},{2},{1,2}}, the

relation on the powerset of A: ={(,),(,{1}),(,{2}),(,{1,2}), ({1},{1}),({1},{1,2}),({2},{2}),({2},{1,2}),({1,2},{1,2})}

Example: Show that the inclusion relation is a partial order on the power set of a set A

Proof:Reflexive: for any XP(A), XX.Antisymmetric: For any X,Y P(A), if

XY and YX, then X=YTransitive: For any X,Y, and ZP(A), if

XY and YZ, then XZ?

The relation < on Z is not a partial order, since it is not reflexive

and is related,{1} and {1,2} is related,{2} and {1,2} is related ， but {1} and {2}

is not related, incomparableRelated: comparablenot related: incomparable

Definition 2.22 ： The elements a and b of a poset (A, ) are called comparable if ≼either a≼b or b≼a. When a and b are elements of A such that neither a≼b nor b≼a, a and b are called incomparable.

≦ The relation on R≦ ， For any x,yR, or x y, or y x≦ ≦ ，thus x and y is comparabletotally order

Definition 2.23 ： If (A, ) is a poset and ≼every elements of A are comparable, A is called a totally ordered or linearly ordered set, and is called a total order ≼or linear order. A totally ordered set is also called a chain

The relation on Z is a total order. The ≦relation | on Z is not a total order. The relation on the power of a set A is not a total order.

2 ． Hasse DiagramsHasse DiagramsDigraph: predigestion (1) partial order is reflexive ，aRa ，We shall delete all loop from the digraph (2) Because a partial order is transitive ， We

do not have to show those edges that must be present because of transitivity.

(3)If we assume that all edges are are pointed “upword”, we do not have to show the directions of the edges.

Hasse Diagrams

The relation on the power of a set AP(A)={,{1},{2},{1,2}}Example: A={2, 3, 6, 12, 24, 36}, (A, |)A={1, 2, 3, 4, 5, 6},(A, )≦

3.Extremal elements of partially ordered setsDefinition 2.24: Let (A, ) is a poset. An ≼

elements aA is called a maximal element( 极大

元 ) of A if there is no elements c in A such that a≺c. An elements bA is called a minimal element ( 极小元 ) of A if there is no elements c in A such that c≺b.

Example ： A1={1,2,3,4,5,6},(A1,)

1 is a minimal element of A1

6 is a maximal element of A1

(A1,|)

1 is a minimal element of A1.

As these example shows, a poset can have more than one maximal element and more than one minimal element.

Definition 2.25: Let (A, ) is a poset. An ≼elements aA is called a greatest element ( 最大元 ) of A if x a for all x≼ A. An elements aA is called a least element ( 最小元 ) of A if a x for all ≼xA.

Note: difference between greatest element and maximal element

Example ： A1={1,2,3,4,5,6},(A1,)1 is the least element of A1. 6 is the greatest element of A1

(A1,|)1 is the least of A1. There is no greatest element.

A2={2,3,6,12,24,36},(A2,|)

There is no greatest element. There is no least element.

Definition 2.26: Let (A, ) is a poset, and ≼BA. An element aA is called an upper bound ( 上界 ) of B if b≼a for all bB. An element aA is called a lower bound ( 下

界 ) of B if a≼b for all bB.Example: A2={2,3,6,12,24,36},(A2,|)

P={2,3,6}, all upper bounds of P are P has no lower bounds.

Definition 2.27: Let (A, ) is a poset, and ≼BA. An element aA is called a least upper bound ( 最小上界 ) of B, (LUB(B)), if a is an upper bound of B and a≼a’, whenever a’ is an upper bound of B. An element aA is called a greastest lower bound ( 最大下界 ) of B, (GLB(B)), if a is a lower bound of B and a’≼a, whenever a’ is an lower bound of B.

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