Question

3. (a Draw a diagram to represent the | (divides) partial order on the set {1, 2, 3, 4, 5, 6 7,8,9, 10, (b) Identify all minimal, minimum, maximal, and maximum elements in the diagram

Answer 1

Definitions:-

Suppose set is P.

Minimal element:- An element a $\epsilon$P is minimal if there is no x $\epsilon$ P such that a<x. In short these are elements that have no tails in hasse diagram.

Maximal element:- An element a$\epsilon$ P is maximal if there is no x $\epsilon$ P such that x < a. In short,these are elements that have no horns in hasse diagram.

Maximum element:- An element a $\epsilon$ P is greatest or maximum if x <= a for all x $\epsilon$ P.(note <= is relation).In short if maximal set has only one element then that element becomes maximum else not .also maximum is always unique.

Minimum element :- An element a$\epsilon$ P is least or minimal if a <= x for all x $\epsilon$ P. in short,if minimal set list has only one element then that element becomes minimum element.

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