Question: 1. For each of the sets below and the corresponding binary relation, prove that the relation is binary relation and find the quotient set.

(a) Let A={1,2,3,4,…} be the set of natural numbers. Consider the binary relation R on A defined by: for all n,m∈A, (n,m)∈R if, and only if, their difference n−m is divisible by 10.

So, for this question, I already verified that it is reflexive, symmetric, and transitive. However, I am not sure how to begin to find the quotient set... Won't there be infinity equivalence classes? 

Can someone please explain to me how to approach this?

Thanks!

Answer: 
Let's look at the class of $0$ :
$$0^\equiv = \{\dots; -20; -10; 0, 10 ; 20 ;\dots\}$$
Now look at the class of $7$ :
$$7^\equiv = \{\dots; -13; -3; 7, 17 ; 27 ;\dots\}$$

Each class is infinite, but there will be exactly 10 equivalence classes. They correspond to the different remainders you can get with an Euclidean division by 10.

In other words, $m\equiv n \Longleftrightarrow m \operatorname{Mod}10 = n \operatorname{Mod} 10$.
One recipe to visualize equivalence classes is to pick a generic natural number and determine which other numbers are equivalent (considered "equal" in that sense) to it. Take $2$, e.g., then the numbers $12$, $22$, $32$, $42$ and so on are equivalent to $2$. Hence $[2]$, the equivalence class of $2$, is
$$[2]=\{2,12,22,32,42,52, \dots\}.$$