I am having problems trying to picture what this relation of ordered pairs 'looks' like:
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b),(c, d)) ∈ R if and only if ad = bc.
I am studying relations and set theory but none of the questions were presented in this manner yet.
How i picture this relation is:
$$ \begin{bmatrix} (a,a) & (b,a) & (c,a) & (d,a) \\ (a,b) & (b,b) & (c,b) & (d,b) \\ (a,c) & (b,c) & (c,c) & (d,c) \\ (a,d) & (b,d) & (c,d) & (d,d) \\ \end{bmatrix} $$
Is this correct?
If I understand what you mean by "picture": To represent a general relation on a set $X$ with a diagram, you'd normally draw a subset $R$ of the Cartesian product $X \times X$. In your example, $X$ is the set of ordered pairs of positive integers, while your diagram only depicts $X$ itself.
To picture the relation $R$ in this sense, you'd need to start with a diagram of the "positive cone in the four-dimensional integer lattice", i.e., the set of ordered pairs of ordered pairs $\bigl((a, b), (c, d)\bigr)$ with $a$, $b$, $c$, and $d$ positive integers.
If $R$ is an equivalence relation, however, you can draw the set $X$ (as you've done) and then label elements of $X$ (e.g., with colored dots) in a way that gives distinct labels to members of distinct equivalence classes. (As the commenters suggest, your $R$ is an equivalence relation, though a proof of this may need to be supplied.)