# List all maximal and minimal elements of the partial order R = {(a,a), (b,b), (c,c), (a,c)}

I know what the definitions of maximal and minimal elements are but I'm not sure how to apply them in this case. Any help would be great.

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## 1 Answer

If you draw a picture of this partial order, you get this:

            c
|
|     b
|
a


The only element with a strictly smaller element is $c$, so $c$ is the only non-minimal element; $a$ and $b$ are minimal, because there is no element strictly smaller than either of them.

Can you tell now what the maximal elements are?

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I'm still a bit confused about why the picture looks like that. –  zeqof Nov 7 '12 at 8:21
Let me write $x\preceq y$ to mean that $\langle x,y\rangle\in R$; it helps the intuition. The ordered pairs $\langle a,a\rangle,\langle b,b\rangle$, and $\langle c,c\rangle$ don’t really tell you anything; they’re just $a\preceq a,b\preceq b$, and $c\preceq c$, which are required since the partial order must be reflexive. The only pair that actually tells you something non-trivial is $\langle a,c\rangle$, or $a\preceq c$; and since $a\ne c$, this actually tells you that $a\prec c$. In this partial order $a$ is smaller than $c$, and that’s the only non-trivial ordering, just as in the diagram. –  Brian M. Scott Nov 7 '12 at 8:26
Right. That clears it up. Thank you. So then a and b are the minimal elements and c is the only maximal element? –  zeqof Nov 7 '12 at 8:30
Sorry, I meant that a and b are minimal and c and b are maximal. Is that right? –  zeqof Nov 7 '12 at 8:42
@zeqof: Yes, that’s right: $b$ is both, $a$ is minimal, and $c$ is maximal. –  Brian M. Scott Nov 7 '12 at 8:44