I am currently struggling to understand how to determine parts b, c and d of this problem. I think I understand the definition of minimal and maximal elements, but what about in this case with multiple conditions? I.E. for this set $X = \{0,1\}$, and condition (i), $0$ satisfies being a partial order of all other elements, but for condition (ii) a,b,c and d could all be $0$, and so for condition (ii) wouldn't $0$ not be a partial order of itself? What am I missing or misunderstanding? Thanks in advance to those who answer!
I think you are misunderstanding the setup of the problem.
The set $A$ consists of the ordered pairs $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$.
The partial order relates two such pairs. For example, $(0,0) \mathscr{R} (1,0)$ because $a=0$ and $c=1$ and $a<c$.