Find all the symmetries of the $ℤ\subset ℝ$. Find all the symmetries of the $ℤ\subset ℝ$.
I'm not sure what is meant with this. 
My frist thought was that every bijection $ℤ→ℤ$ is a symmetry of $ℤ$. 
My second thought was that if I look at $ℤ$ as point on the real line, then many bijections would screw up the distance between points. Then I would say that the set of symmetries contains all the translation: $x↦x+a$ and the reflections in a point $a∈ℤ$, which gives, $x↦a-(x-a)$.

Both of you thoughts are correct, but in different contexts. If we regard $\mathbb{Z}$ as just a set with no extra structure then the symmetries of that set are just the bijections, (as in this case, we are thinking as $\mathbb{Z}$ as a bag of points). If on the other hand, we add the extra structure of the distances, then the symmetries are more restricted and the symmetries are those maps that preserve distance.
