Informally describe the set $A =\{n : n \text{ is an integer and } n = n + 1\}$
Just a quick set theory exercise. At first my reaction was: there is no integer that satisfies this description, i.e., $A = \emptyset$. But on second thought, the set may be non null if it was
$$A = \mathbb{Z}_1$$
because $n \equiv k \equiv 0 \bmod 1 $ for any integer $n$. Maybe this question was designed to be vague (it's from a Theory of Computation text). Or maybe I'm taking too much freedom with my interpretation of equality. What do you think?
Your first thought is correct. There are no integers $n$ with $n=n+1$.
Your second thought is interesting, but I wouldn't say that you're taking liberties with your interpretation of equality so much as with your interpretation of integer.
When we say integer, we mean an element of $\Bbb{Z}$ with its usual ring structure (i.e. with its usual 0, 1, and addition, and multiplication, (and if you'd like to think of it that way, with its usual equality)), not an element of $\Bbb{Z}/1\Bbb{Z}$ or $\Bbb{Z}/n\Bbb{Z}$ for any $n$.