Let $S = \{ u,v,w \} $ List all equivalence relations on S. How many of these are also partial orders?
Well I found five total equivalences classes but right now I'm confused as to which are partial orders. The confusion is because I'm thinking in transitive terms. For example, one equivalence relation I found is:
$$\{ (u,u),(v,v),(w,w),(u,v),(v,u) \}$$
Anti symmetric means that $(u,v) \land (v,u) \rightarrow u=v$ but $u$ and $v$ are two different variables and so they're not the same. So this means that they're antisymmetric? I don't know why I'm stuck here.