Question

Let `R={(a,b):a,b inZ" and "(a-b)" is even"}.`
Then, show that `R` is an equivalence relation on `Z`.

Answer 1

here ,R satisfies the following properties :
(i) Reflexivity
Let a be an arbitrary element of Z.
then ,(a-a)=0 , which is even
`therefore (a,a)in R Aaa in Z.`
So , R is reflexive.
(ii) Symmety
Let a,b `in ` Z such that (a,b) `in R` . then,
`(a,b) in Rimplies (A-b)` is even
`implies -(a-b)`is even
`implies (b-a) ` is even
`implies (b,a) in R.`
`therefore ` R is symmetric .
(iii) Transitivity
Let `a,b,c in Z`such that ` (a,b) in R and (b,c) in R.` then ,
`(a,b)in R and (b,c) in R`
`implies (a-b)` is even and (b-c) is even
`implies {(a-b)+(b-c)}` is even
`implies (a-c)` is even
`therefore ` R is transitive .
thus ,R is reflexive , symmetric and transitive .
hence ,R is an equivalence relation in Z.