Thread: Equivalence classes

This is a question from my professor that I couldnt understand.
Let X=R, x~y if x-y is a subset of Z(integers).
Describe the set of equivalence classes.
He said it's just a circle.
I have no idea what he meant.
Anyone can understand what he meant?
THanks.

Originally Posted by ninano1205

This is a question from my professor that I couldnt understand.
Let X=R, x~y if x-y is a subset of Z(integers).
Describe the set of equivalence classes.

Just what are you saying, "x-y is a subset of Z".
Are you saying that the difference is a integer?

is integers?? right?

Originally Posted by ninano1205

is integers?? right?

I thought as much. In that case, the mention of circle has no relevance.
As an aside, this fits into a different problem that I have worked recently.
Now one equivalence class is simply the set of integers: .

Do you know about the floor function?
is the greatest integer not exceeding .
We can show that this function is well defined on having the property that:
.

Now back to your question. Define a function on as (decimal part) .
Take note that .
For the general equivalence classes: .