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Consider the poset $( \{3,5,9,15,24,45 \}, \mid).$

Which of the following is correct  for the given poset ?

1. There exist a greatest element and a least element
2. There exist a greatest element but not a least element
3. There exist a least element but not a greatest element
4. There does not exist a greatest element and a least element

edited | 772 views

There are two maximal elements $24$ and $45$.

There are two minimal elements $5$ and $3$.

So there is no greatest and least element.

$\therefore$ Option $4.$ is correct.

by Boss (23.8k points)
edited
0
Can there ever be two greatest elements?
+2
No....because we can't compare  them. there can be many maximal elements but only one maximum element.

1-We can not choose here greatest element because two maximal element(24,45 are at same level in Hasse diagram) are there.                                                                                                                                                                                                                              2-We can not choose here least element because two minimal element(3,5 are at same level in Hasse diagram) are there.                                                                                                                                                                                                                                          So: Option 4 is correct.

by (263 points)
edited
–1 vote
C is correct answer because there exists LCM(A,B) FOR all a,b belongs to the set
by (47 points)

+1 vote