Describe the set of relations on $\mathbb{Z}$ which are both symmetric and anti-symmetric. Hint: this set is infinite and contains one relation with which you are already familiar.
I know that this is clearly talking about the equality relation, but what I am confused about is what does a set of relations mean if there is only one relation? Wouldn't the relation just be $S=\{(x,y)\in\mathbb{Z}^2:x=y\}$? Or does the set of relations mean the powerset of $S$?
On
You are correct that $S$ is the relation denoting equality. However note that any subset of $S$ is also a relation that is symmetric and antisymmetric. The idea here is that we do not need to define all the integers to be equal to each other.
Thus the answer you are looking for is the set of all subsets of $S$ (which is infinite).
On
The identity/equality relation is both anti-symmetric and symmetric. However it is not the only such. The empty relation is also both anti-symmetric and symmetric. There are many others.
You wish to describe all relations (on the integers) which have the properties of both symmetry and anti-symmetry. What are these properties and identify how may a relation possess both?
A relation $R$ on set $\Bbb Z$ is symmetric when ....
A relation $R$ on set $\Bbb Z$ is anti-symmetric when ....
Therefore a relation $R$ on set $\Bbb Z$ is both anti-symmetric and symmetric when ....
You're correct, it is the power set of $S$, namely: $$ \mathcal P(S) = \{\{(x, x) \mid x \in A\} \mid A \subseteq \mathbb Z\} $$