We've got these relations and have to decide if the orders are strong, weak or nothing. I am very unsure how to check this and got to these conclusions:
$\{⟨a,b⟩,⟨a,c⟩,⟨a,d⟩,⟨b,c⟩,⟨a,a⟩,⟨b,b⟩,⟨c,c⟩,⟨d,d⟩,⟨c,b⟩\}$ - nothing
$\{⟨b,a⟩,⟨c,b⟩,⟨c,a⟩\}$ - weak
$\{⟨d,c⟩,⟨d,b⟩,⟨d,a⟩,⟨c,a⟩,⟨a,b⟩\}$ - strong
Am I correct here or do I misunderstand it? Is there are easy way to check this? I'd really appreciate any help.
The first would be a weak order if we added either $\langle d,b\rangle$ and $\langle d,c\rangle$, or $\langle b,d\rangle$ and $\langle c,d\rangle$, but as it stands $d$ is not comparable with the $\{c,d\}$ pair, so it’s a preorder, but not a total preorder and hence not a weak order. Since it’s not a weak order, it certainly cannot be a strong order. Of the three choices available to you it is indeed nothing.
The second is a strict strong order: there are no violations of antisymmetry, as there can be for a weak order.
The third is not transitive: $\langle c,a\rangle$ and $\langle a,b\rangle$ are in the set, but $\langle c,b\rangle$ is not. Thus, this is not an order at all.
Note: There is considerable variation in terminology, and it’s possible that the definitions to which I’m accustomed are not the same as the ones that you’re using. It would be a good idea to add those definitions to your question.