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The question is, "Which of these relations on$\{0,1,2,3\}$ are partial orderings? Determine the properties of a partial ordering that the others lack."

The only two I had trouble with were:

$\{(0,0), (1,1), (2,0), (2,2), (2,3), (3,2), (3,3)\}$


$\{(0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0)(2,2), (3,3)\}$

For the both, I supposed that they were transitive, but the answer key says otherwise.

I honestly can't find the missing elements, the ones that make it not transitive.

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Have you drawn these? – Berci Nov 12 '12 at 17:34
Do you mean with digraphs? – Mack Nov 12 '12 at 17:37
Well. Yes. What else? $4$ points and connect them if they are in the given relation. Instead of loops, you can just mark anyhow the reflexive pointpairs of the form $(x,x)$. – Berci Nov 12 '12 at 17:38
up vote 3 down vote accepted

The first is not transitive: $3\sim 2$ and $2\sim 0$, but $3\not\sim 0$. It’s also not antisymmetric: you have both $2\sim 3$ and $3\sim 2$, even though $2\ne 3$. The second also fails to be antisymmetric ($0\sim 2\sim 0$, but $0\ne 2$, and $0\sim 1\sim 0$, but $0\ne 1$) or transitive: $2\sim 0$ and $2\sim 1$, but $2\not\sim 1$.

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For transitivity:

  1. $(3,0)$ is missing though $(3,2)$ and $(2,0)$ are there.
  2. $(2,1)$ is missing though $(2,0)$ and $(0,1)$ are present.
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