2.6 Equivalence relation 1. Equivalence relation Definition 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>l Show that congruence modulo m is an equivalence relation. R=(a, b )la=b mod m) Proof: (Reflexive (for any aEZ, aRa?) 2)symmetric (for any arb, bRa?) transitive(for arb, bRc, arc?)
2.6 Equivalence Relation 1.Equivalence relation Definition 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 classes partition Definition 2. 19: A partition or quotient set of a nonempty set A is a collection II of nonempty subsets ofa such that (1) DEach element of a belongs to one of the sets in II. (2)IfAi and Ai are distinct elements of Il, then A∩A: The sets in l 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,(},V={a,b},{b,},W={a,b},{a,c},(C}, nfinite
2.Equivalence classes partition Definition 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 xEZ, or x=0 mod 2,or x=I mod 2, le or x∈E,orx∈O. AndE∩O= E and o RE, O is a partition of Z
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 lairs 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}= ={,-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]l={…,-n-1,1,n-1,2n-1,3n-1,} A partition or quotient set of Z, Z/R={01,1l…,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]}