The question I am trying to solve is below. I have proven it is an order but am unsure how to draw the diagram for it. Can someone point me in the right direction?
Let $A = \{1, 2, 3, 4\}$, and let $R$ be a binary relation on $A \times A$ given by: $((a, b),(c, d)) \in R$ if and only if $a$ divides $c$ and $b$ divides $d$.
What I have so far, on the right track?
Second Level: $(1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)$
Bottom Level: $(1,1)$
As was noted in the comments, you have far too much in your second level. For instance, $2\mid 4$ and $1\mid 1$, so $\langle 2,1\rangle$ must be below $\langle 4,1\rangle$. On the other hand, we know that there’s nothing between them, because there is no integer $n\{1,2,3,4\}$ such that $2\mid n$, $n\mid 4$, and $n$ is neither $2$ nor $4$.
Here are the bottom three levels; can you finish it from here?