1. ## Equivalence relation

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.

2. ## Re: Equivalence relation

your proof would read better if you use x~y for xRy instead of "=". you also have a typo in your proof for symmetry, it should read y~x, not y~y.

your listing of the elements of the equivalence classes would be more convincing with the inclusion of negative integers as well:

[0] = {...-8,-4,0,4,8,....}
[1] = ?
etc.

for example, what is [-3]?