Transcription of Congruence and Congruence Classes
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Congruence and Congruence Classes
LECTURE 11 Congruence and Congruence relation on a setSis a rule or test applicable to pairs of elementsofSsuch that(i)a a , a S(reflexive property)(ii)a b b a(symmetric property)(iii)a bandb c a c(transitive property).You should think of an equivalence relation as a generalization of the notion of equality. Indeed, the usualnotion of equality among the set of integers is an example of an equivalence relation. The next definitionyields another example of an equivalence , b, n Zwithn >0. Thenaiscongruent tobmodulon;a b(modn)provided thatndividesa 5 (mod 6)The following theorem tells us that the notion of Congruence defined above is an equivalence relation on theset of a positive integer. For alla, b, c Z(i)a a(modn)(ii)a b(modn) b a(modn)(iii)a b(modn)andb c(modn) a c(modn).Proof.(i)a a= 0 andn|0, hencea a(modn).(ii)a b(modn) means thata b=nkfor somek Z.
Congruence and Congruence Classes Definition 11.1. An equivalence relation ~ on a set S is a rule or test applicable to pairs of elements of S such that (i) a ˘a ; 8a 2S (re exive property) (ii) a ˘b ) b ˘a (symmetric property) (iii) a ˘b and b ˘c ) a ˘c (transitive property) :
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