Question

Let I be the set of all interger and R be the relation on I defined by aRb iff (a +b) is an even integer for all a,b belongs to R prove that it is a eqivalence relation

Answer 1

Answer:

1. Reflexive clearly, (a+a) = 2a i.e even hence aRa it is reflexive
2. Symmetric let aRb = (a+b)=2m

=( b+a) = 2m

= bRa

=it is symmetric

3. Transitive let tab and Bec

= (a+b)=2m and (b+c)=2 n

= (a+c) =2(m+n)

aRc i.e transitive

hence it is an equivalence relations