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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 × A given by: ((a, b),(c, d)) ∈ 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)

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  • $\begingroup$ In the second level just write $(1,2),(1,3),(2,1)$ and $(3,1).$ Don't write $(2,2),$ for example, since you have $(1,2)$ and $(2,1)$ in the middle. $\endgroup$
    – mfl
    Oct 18, 2014 at 23:16

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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?

enter image description here

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  • $\begingroup$ Thanks I see it now! I can solve it from here :) $\endgroup$
    – user3362196
    Oct 19, 2014 at 2:43
  • $\begingroup$ @user3362196: You’re welcome! $\endgroup$
    – Brian M. Scott
    Oct 19, 2014 at 2:45

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