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, $x↦a-(x-a)$.

It probably means the latter, in this case. Note that we can also reflect over the midpoint of two (successive) points in $\Bbb Z$. Otherwise, you've got it covered.

**Addendum** : It might be fun for you to prove that every such symmetry of $\Bbb Z$ can be obtained as a composition of finitely-many instances of the two maps $x\mapsto x+1$ and $x\mapsto-x$.