Find out the maximal elements of a poset, Mathematics

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Refer the poset  ({1}, {2}, {4}, {1,2}, {1,4}, {2,4}, {3,4}, {1,3,4}, {2,3,4}, ≤ ).

(i)  Find out the maximal elements.

(ii)  Find out the minimal elements.

(iii)  Is there a least element.

(iv)  Find out the least upper bound of {{2}, {4}} if it exists.   

Ans:  (i) Maximal element in a poset is illustrated as element that is not succeeded by any type of other element in the poset. The maximal elements are {1, 2}, {1, 3, 4} and {2, 3, 4}

(ii) Maximal element in a poset is illustrated as element that is not preceded by any other element in the poset. The minimal elements are {1}, {2} and {3}

(iii) There is no least element in the poset, like there exist no element x like that x precede every element of the poset. For instance neither {1} precede {2} nor {2} precede {1}.

(iv)   The upper bound of {{2}, {4}} are {2, 4} and {2, 3, 4}. The least of the upper bounds is {2, 4}.


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