I have the following question for homework
I'm unsure what the first part of the question refers to. I'm assuming its asking for the size of the power set of A but I haven't seen any mention of power sets regarding relations.
The second part refers to the size of the Cartesian product A^2(A * A).
The third and fourth part I don't recall seeing in my text book and I can't seem to find any mention of "collection of relations" online.
On
$C$ is not mentioned in the first part. You are correct that it is just looking for the size of the power set of $A$ and $8$ is correct. The third part is asking you to find how many different relations there are on $A$. You should review the definition of a relation to find the size of this set. Since a relation is just a set of ordered pairs, one can certainly talk of the Cartesian product of two of these sets.
Recall a relation on a set $A$ is a subset of $A \times A$. Note that since $|A|=3$ then $|A \times A| = 9$. Letting $C$ be the collection relations on $A$, we then would have:
\begin{array} \ |C| &= \text{ amount of subsets of $A \times A$ } \\ &= | \mathcal{P}(A\times A)| \\ \end{array}