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$\begingroup$

When asked a question like this:

Give an example of a nonempty partially ordered set (S, R) that does not have incomparable elements. Draw the Hasse diagram for this partially ordered set

would this be a correct answer:

$S =$ {$1, 2, 4, 6$}

$R =${$(1, 1), (1, 2), (1, 4), (2, 2), (4, 2),(2,6), (4, 4), (6, 6)$}

$\endgroup$
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  • $\begingroup$ Apparently not. $4$ and $6$ appear to be incomparable. $\endgroup$
    – Matt Samuel
    May 19, 2015 at 0:11
  • $\begingroup$ @MattSamuel what if I just did S= {1,2,4,8}? $\endgroup$
    – Csci319
    May 19, 2015 at 0:12
  • $\begingroup$ There's a bit of a deeper problem, notice that your relation is not even transitive, so it's not a partial ordering. $\endgroup$
    – Matt Samuel
    May 19, 2015 at 0:13
  • $\begingroup$ take any linear order, say $S=\{0,1\}$, $R=\{(0,0),(0,1),(1,1)\}$, or $S=\{0\}, R=\{(0,0)\}$. $\endgroup$
    – yoyo
    May 19, 2015 at 0:14
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    $\begingroup$ "Partially ordered set that does not have incomparable elements" is just a long-winded way of saying "totally ordered set". $\endgroup$
    – hmakholm left over Monica
    May 19, 2015 at 0:15

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