Question

Let `n` be a positive integer. Prove that the relation R on the set Z of all integers numbers defined by `(x , y) in R iff x-y` is divisible by `n ,` is an equivalence relation on Z.

Answer 1

Here, `R = {(x,y):x,y in R and (x-y)` is divisible by `n}`
For all `a in R`,
`=> (a-a) =0` and `0` is divisible by `n`.
`:. (a,a) in R`.
`:. R` is reflexive.

Since in `R` for every `(a,b) in R`
`=> (a-b)` is divisible by `n`.
`=> (-(b-a))` is divisible by `n`.
`=> (b-a)` is also divisble by `n`.
`:. (b,a) in R`.
`:. R` is symmetric.

Since `(a,b) in R and (b,c) in R`
`=> (a-b)` is divisible by `n` &  `(b-c)` is divisible by `n`.
`=> (a-b+(b-c))` is divisible by `n`.
`=> (a-c)` is divisible by `n`.
`:. (a,c) in R`.
`:. R` is transitive.
As `R` is reflexive, symmetric and transitive, `R` is an equivalence relation.