• T is defined on the set {a, b, c, d, e},
• aTc, bTc, cTd, eTb and e not T a,
• T is a partial order relation.
Draw a Hasse diagram for every possible relation that T could be and, for each, write down whether it is a total order relation.
The problem that I am having is where to put the letters on the diagram since I know that in Hasse diagrams, some elements have higher importance than others. When I assume that the relationship is less than or equal to, i get 1 diagram with d at the top and e at the bottom. If I were to change the relationship to greater than or equal to, then would I just flip the entire thing? Is my thinking on the right track?
You can think simply in terms of directed graphs. Plot your 5 vertices on a circle, as vertices of a regular pentagon.
Then being a Hasse diagram means being both acyclic (no oriented cycle) and transitively reduced (no shortcut-edge). This should be easy to see on the picture. You can try adding edges and see what happens to oriented cycles and shortcuts.
Only after you know that you have a Hasse diagram, you can choose if you want minimum elements at top or at bottom.