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, $x↦a-(x-a)$.