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 lars When on/v 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}= ={,-3,-1,1,3,}=|3=-1l-3}= the partition of ZIL/R=o1
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,} I={…,-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
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 a∈A,a∈|al}; (2)IfaR b, then a=bl; 63)Fora,b∈A,If(a,b)gR, then a∩bl=; (4)∪a=A Proof: (1)For any aEA, aRa? (2)For a, bEA, arb, as?b,bc?a For any xelal,xe?b when arb,i.exR b for any xE, xE? a when arb, i,exRa 3Fora,b∈A,If(a2b)gR, then a∩b]= Reduction to absurdity Suppose a]∩bl≠, Then there exists x∈a]∩b (4)
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 absurdity Suppose [a]∩[b]≠, Then there exists x[a]∩[b]. (4) a A a A = (4)[ ]
The equivalence classes of an equivalence relation on a set form a partition of the set Equivalence relation =partition Example: Let A=(1, 2, 3, 43, 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
The equivalence classes of an equivalence relation on a set form a partition of the set. Equivalence relation partition Example：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 IA1,A2y.,An be a partition of a nonempty set A. Let r be a relation on A, and arb if only if there exists A Ell s.t. a,b∈A ie.R=(A1×A1)∪U(A2×A2)U….∪(An×An) R is an equivalence relation on A Theorem 2.12: Given a partition AiEZ of the set A, there is an equivalence relation R that has the set A; IEZ, as its equivalence classes
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 )∪(A2A2 )∪…∪(AnAn ) R is an equivalence relation on A Theorem 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 Ii-a, b,,c be a partition of A=(a,b, c). Equivalence relation r-
Example: Let ={{a,b},{c}} be a partition of A={a,b,c}. Equivalence relation R=?
2.7 Partial order relations 1. Partially ordered sets Definition 2.21: A relation 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 denoted that (a, bER. Note that the symbol s 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
2.7 Partial order relations 1.Partially ordered sets Definition 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 ab
The relation on R: The relation on Z: the relation c on P(A) partial order, R, s ),(Z, /),(P(A),C are partially ordered sets Example:LetA={1,2},P(A)={,{1}2{2},{1,2},the relation on the powerset of A: ={(,),(,{1}),(,{2}),(,{1,2}) (1},{1}),({1},{1,2})({2},{2})2({(2},{1,2}),({1,2},{1,2})}
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 c is a partial order on the power set of a set a Proof: Reflexive: for any X∈P(△),XcX Antisymmetric: For any X,Y EP(A), if XcY and Yax. then Xy Transitive: For any X,Y, and ZEP(A), if XcY and Ycl, then Xcz?
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=Y Transitive: 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 o and o is related 1 and ( 1, 2) is related, 12 and ( 1, 2 is related, but 1 and (2 is not related, incomparable Related: comparable not related: incomparable
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, incomparable Related: comparable not related: incomparable