# 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?

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If you don’t see the answer right away, note that there are only $15$ elements in that partial order, so it’s no great labor to write them all out. – Brian M. Scott Nov 13 '12 at 0:16
Also, it's always good to work from definitions, once you've written them all out (the elements): what does it mean to be a maximal, minimal, least, greatest...element? – amWhy Nov 13 '12 at 0:18
Sorry, but can you list all the pairs in the relation? I want to make sure mine is correct. It will be so much helpful to me! – Aaron Nov 13 '12 at 0:19
That’s not a reasonable request, I’m afraid: there are some $50$ pairs. – Brian M. Scott Nov 13 '12 at 0:26
Ok! that's fine I will try to figure this out – Aaron Nov 13 '12 at 0:32

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?

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{a},{b},{c},{d} are the minimal elements of R with no least element. – Aaron Nov 13 '12 at 0:33
Also {a,b,c,d} is the maximum element with that being the greatest element. – Aaron Nov 13 '12 at 0:33
Is this correct? – Aaron Nov 13 '12 at 0:34
Absolutely right, @Aaron! – Cameron Buie Nov 13 '12 at 0:35
@Aaron: You’ve got it: that is indeed correct. – Brian M. Scott Nov 13 '12 at 0:35

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 $R$?

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