We've started studying relations and I have no idea what I'm doing. Is this supposed to be the way to represent my answers?
a.
The relation is all possible relations such that the sum of a,b is equal to the sum of c,d. So in order to avoid double counting we basically need to find a way to reach all numbers in [2,8].
$\left\{\left[1,1\right]_{R}\ ,\left[1,2\right]_{R},\left[2,2\right]_{R},\left[2,3\right]_{R},\left[3,3\right]_{R},\left[3,4\right]_{R},\left[4,4\right]_{R}\right\}$
b.
The relation is all possible relations such that the sum of a1,a2,a3 is equal to the sum of b1,b2,b3. So in order to avoid double counting we basically need to find a way to reach all numbers in [0,3].
$\left\{\left[0,0,0\right]_{R}\ ,\left[1,0,0\right]_{R},\left[1,1,0\right]_{R},\left[1,1,1\right]_{R}\right\}$
I think you're almost there. I believe they want some explanation for why these are the answers (perhaps for $a$ you can say that two pairs are equivalent if they have the same sum, so we only have to find one representative for each sum) I mean, you need to explain what you mean by "find a way to reach all [2,8]".
However I believe you have a mistake. The equivalence class of $[2,2]$ is equal to the equivalent class of $[1,3]$ because $2+2=1+3$.