For example
I am not sure how to approach this type of problem. I know that the equivalence classes partition $A$.
Suppose $[a]= \{1,4,5\}$, $[b]=\{2,6\}$ and $[c]= \{3\}$. $[a]\bigcap[b]= \emptyset$ and $[b]\bigcap [c]=\emptyset$, but I have no idea how to find the relation as a subset of $A \times A$.
$A \times A$ is the Cartesian product of $A$ with itself. On a smaller set, it might be easier to see what this means. So, let $B = \{1,2,3\}$
$B \times B = \{ (1,1), (1,2), (1,3),(2,1), (2,2), (2,3),(3,1),(3,2),(3,3)\}$
So, a subset of this would be $ \{(1,1),(2,2),(3,3)\}$ for instance.
For your question, we know the equivalence classes, which means that if two elements $a,b$ are in the same equivalence class, then $aRb$ and $bRa$. Note: $a$ can equal $b$, so you always have $aRa$ in an equivalence relation.
So, for the equivalence class $\{2,6\}$, you will have $\{(2,2),(2,6),(6,2),(6,6)\}$ as part of the relation.
You simply need to do the same operation for the rest of the equivalence to define your relation $R = \{(1,1),(1,4),(1,5),...\}$. They are simply asking for you to write your answer as a set of points as opposed to listing a function or using $aRb$ notation.