Partial Order least and greatest elements Prove that the following relation on the set of all nonempty subsets of $\{a,b,c,d\}$ is an order, draw its diagram, find all the maximal, minimal, least and greatest elements:

$(x,y)\in R$ if and only if $x$ is a subset of $y$

How do I determine the maximal, minimal, least and greastest elements?
 A: Minimal/maximal means there aren't any elements that are less/greater. Least/greatest means less/greater than all the others.
If a partial order has a least/greatest element, then it is the unique minimal/maximal element. A minimal/maximal element is only the least/greatest if it's less/greater than or equal to all the elements. In particular, if there's more than one minimal/maximal element, then there is no least/greatest element.
What do these things mean in this context, given the definition of $R$?
A: HINTS: Use the definitions. 
What does it mean to say that $x$ is a minimal element in this partial order? It means that there is no $y$ such that $y\subsetneqq x$. Is that true of the element $\{a,c\}$, for instance? No, because $\{a\}\subsetneqq\{a,c\}$. Therefore $\{a,c\}$ cannot be a minimal element. 
Similarly, $x$ is maximal if there is no $y$ such that $x\subsetneqq y$. Thus, $\{a,c\}$ is also not maximal, because $\{a,c\}\subsetneqq\{a,b,c\}$.
Finally, $x$ is the greatest element if every $y$ in the order satisfies $y\subseteq x$; is there an $x$ like that?
