Prove equivalence realtion and list the equivalence classes

Define xRy if x=y (mod4) if x=y (mod 4) then 4/x-y

Reflexive: Suppose x in R, then x-x=0, which is divisible by 4, so (mod4) is reflexive.

Symmetric: Suppose xy in R, Then (x-y) =4K for some k, then y-x= -4K= 4 (-k)

So y=y (mod4) and (mod4) is symmetric.

Transitive: Suppose, x,y,z in R with x=y (mod4) and y=z (mod 4).

then x-y=4k and y-z=4m for some k, m in R.

SO x-z= (x-y)+ (y-z) = 4k+4m= 4(k+m). Thus x=z (mod 4) as well (mod4) is transitive.

list of the equivalence classes:

[0] = {0,4,8...}

[1] = {1,5,9...}

[2] = {2,6,10..}

[3] = {3,7,11..}

[4] = [0], [5] =[1]

is this correct?

Thank you.