Theorem 2.11: LetR be an equivalence relation on A. Then (1) For any a∈A,a∈al; (2)IfaR b, then a=bl; 3)Fora,b∈A,If(a2b)R,then[a∩b=; (4)∪a=A Proof: (1)For any aEA, ara (2)For a, bEA, arb, as?b], bc?al For any x∈|a],x∈?[b] when arb,ie,xRb for any x∈b],x∈?| a] when arb,i,e.xRa 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 aA, a[a]; (2)If a R b, then [a]=[b]; (3)For a,bA, If (a,b)R, then [a]∩[b]=; Proof：(1)For any aA，aRa? (2)For a,bA, 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,bA, 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)[ ]