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:
$\{(0,0), (1,1), (2,0), (2,2), (2,3), (3,2), (3,3)\}$
and
$\{(0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0)(2,2), (3,3)\}$
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.
Best Answer
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 $2\not\sim 1$.