Let A be the set of U. S. states. One example of a relation on A is
R = {(s,t) : s = t or s shares a border with t}.
Notice that the domain of R is A, and the range of R is A. Give a different example of a nontrivial relation from A to A whose domain and range are both the set A.
Would using the first letters of states be a proper example?
You seem to be fundamentally misunderstanding something here.
Given a set $S$, a relation $R$ is simply defined as some subset of $S^2$, that is, where $$R=\{\langle s_1, s_2\rangle\in R: s_1\in S, s_2\in S\}$$ $R$ is by definition a relation.
The domain of $R$ (admittedly this isn't a term I haven't heard or seen before, but from intuition and Google this seems to be the most logical definition) is just the set of all left-hand-side elements: $$\{s_1:\exists s_2\in S, \langle s_1, s_2\rangle\in R\}$$
and similarly the range of $R$ is just the set of right-hand-side elements: $$\{s_2:\exists s_1\in S, \langle s_1, s_2\rangle\in R\}$$
So think about your question. What is the domain and range of your specified relation?
