How is a relation between elements inside a set an equivalence relation?

Question

From Rosen's Discrete Mathematics and Its Applications, 3ed, chapter 9 p. 612:

How can they tell $R$ is an equivalence relation right off the bat?

Answer 1

More generally, for any function $f:A\to B$, the relation $\{(a, a') :f(a) =f(a')\}$ is always an equivalence relation.

Prove it in this generality, and find the sets $A,B$ and the function $f$ to apply this for the problem.