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Consider the following statements S1 and S2 :

S1 :

The minimal elements of a poset always form an antichain.

S2 :

The maximal elements of a poset always form an antichain

Which of the following is correct?

Can someone explain these two with examples? Thank you!

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Hasse diagrams for four different posets. Poset D has a disconnected Hasse diagram with two connected components f a ; c ; e g and f b ; d g . in this    In poset ‘A’ maximal element set (yes it is a set) contains just a single element which is ‘a’ means it is the maximum/greatest element. Similarly,minimal element set contains just a single element which is ‘d’ means it is the minimum/least element ” 

 now come to the poset B  which contain minimal elements { e f } and maximal elements { a b   }   in this elemets of maximal set and elements of minimal set belongs to the antichain ( any two distict  element  are incomaparable / no edge between them …..
poset C   mimimal elements {  c h d  } 
                maximal elements { a f  } 

poset D  minimal elements { e d }   

             maximal elements { a b }   

(  their is  diffrenece between “ maximum and miximal elemets “ & “ minimum and minimal element “  if their is only one minimal element in that case it is minimum and  if their is only one maximal element than it is maximum..)

     if their  any mistake please notify me …...thanks 

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