Here is the problem:
Let R be the relation on N given by aRb if and only if 5 divides a-b.
a. Verify that R is an equivalence relation.
b. List the equivalence classes of R as sets. List at least 5 elements of each class; two of these elements should be negative.
I have already verified that R is an equivalence relation, but I am confused on where to start to list the equivalence classes of R as sets. Any explanation on how to do this would be helpful.
Best Answer
The equivalence class of an element $r$ is given by \begin{align*} [r]&=\{x:5\text{ divides }(x-r)\}\\ &=\{x:x-r=5n\}\\ &=\{x:x=5n+r\} \end{align*} So, it is easy to see that $[r]$ is basically the set of all numbers $n$ such that $n\equiv r\pmod 5$. So, the equivalence classes are $[0]$, $[1]$, $[2]$, $[3]$, $[4]$.
Now, try listing the elements.