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Find the minimal and maximal elements of $P^*$?


Determining minimal and maximal elements in products of partial ordersMaximal and maximum (matchings)Question about Hausdorff Maximal principle and antichainSubset of infinite partially-ordered powerset may not have maximal or minimal elementsCountability of minimal and maximal elements.Ordered set and minimal elementDivisibility Relation On the Set $S = 2, 6, 7, 14, 15, 30, 70, 105, 210 $: Hasse Diagram, Maximal, Minimal Elements, Greatest, Least elementsExistence of a countable family of pairwise non-similar partial orders with $2^c$ maximal elements, and no minimal elementsStrange problem about minimal and maximal elements?













1












$begingroup$


Consider the set $1, 2, 3, 4$ and let $P^*:=P(1, 2, 3, 4)-emptyset$ be the set of all subsets of $1, 2, 3, 4$ excepting $emptyset$. Then $P^*$ is ordered via inclusion as follows:



$$Aleq BLeftrightarrow Asubset B.$$



I'm supposed to find the minimal and maximal elements with respect to that order. Can anyone help me?



I believe the minimal elements are the unitary sets $1, 2, 3$ and $4$ and the unique maximal element is $1, 2, 3, 4$. Am I right?










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Yes, that's right.
    $endgroup$
    – Théophile
    yesterday










  • $begingroup$
    Thanks a lot =)
    $endgroup$
    – PtF
    yesterday















1












$begingroup$


Consider the set $1, 2, 3, 4$ and let $P^*:=P(1, 2, 3, 4)-emptyset$ be the set of all subsets of $1, 2, 3, 4$ excepting $emptyset$. Then $P^*$ is ordered via inclusion as follows:



$$Aleq BLeftrightarrow Asubset B.$$



I'm supposed to find the minimal and maximal elements with respect to that order. Can anyone help me?



I believe the minimal elements are the unitary sets $1, 2, 3$ and $4$ and the unique maximal element is $1, 2, 3, 4$. Am I right?










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Yes, that's right.
    $endgroup$
    – Théophile
    yesterday










  • $begingroup$
    Thanks a lot =)
    $endgroup$
    – PtF
    yesterday













1












1








1





$begingroup$


Consider the set $1, 2, 3, 4$ and let $P^*:=P(1, 2, 3, 4)-emptyset$ be the set of all subsets of $1, 2, 3, 4$ excepting $emptyset$. Then $P^*$ is ordered via inclusion as follows:



$$Aleq BLeftrightarrow Asubset B.$$



I'm supposed to find the minimal and maximal elements with respect to that order. Can anyone help me?



I believe the minimal elements are the unitary sets $1, 2, 3$ and $4$ and the unique maximal element is $1, 2, 3, 4$. Am I right?










share|cite|improve this question











$endgroup$




Consider the set $1, 2, 3, 4$ and let $P^*:=P(1, 2, 3, 4)-emptyset$ be the set of all subsets of $1, 2, 3, 4$ excepting $emptyset$. Then $P^*$ is ordered via inclusion as follows:



$$Aleq BLeftrightarrow Asubset B.$$



I'm supposed to find the minimal and maximal elements with respect to that order. Can anyone help me?



I believe the minimal elements are the unitary sets $1, 2, 3$ and $4$ and the unique maximal element is $1, 2, 3, 4$. Am I right?







abstract-algebra order-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday









user26857

39.3k124183




39.3k124183










asked yesterday









PtFPtF

4,04821734




4,04821734







  • 1




    $begingroup$
    Yes, that's right.
    $endgroup$
    – Théophile
    yesterday










  • $begingroup$
    Thanks a lot =)
    $endgroup$
    – PtF
    yesterday












  • 1




    $begingroup$
    Yes, that's right.
    $endgroup$
    – Théophile
    yesterday










  • $begingroup$
    Thanks a lot =)
    $endgroup$
    – PtF
    yesterday







1




1




$begingroup$
Yes, that's right.
$endgroup$
– Théophile
yesterday




$begingroup$
Yes, that's right.
$endgroup$
– Théophile
yesterday












$begingroup$
Thanks a lot =)
$endgroup$
– PtF
yesterday




$begingroup$
Thanks a lot =)
$endgroup$
– PtF
yesterday










0






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oldest

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