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.
List all maximal and minimal elements of the partial order R = {(a,a), (b,b), (c,c), (a,c)}
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1 Answers
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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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0I'm still a bit confused about why the picture looks like that. – 2012-11-07
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0Let 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. – 2012-11-07
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0Right. That clears it up. Thank you. So then a and b are the minimal elements and c is the only maximal element? – 2012-11-07
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0Sorry, I meant that a and b are minimal and c and b are maximal. Is that right? – 2012-11-07
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0@zeqof: Yes, that’s right: $b$ is both, $a$ is minimal, and $c$ is maximal. – 2012-11-07
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0@zeqof: You’re welcome. – 2012-11-07