What does it mean that X is ordered by divisibility when all elements of X are prime numbers? Also doesn't there have to be some partial ordering relation for there to be maximal and minimal elements? Thanks in advance for any clarifying answers.
2026-03-25 04:43:43.1774413823
Let X := {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43} be ordered by divisibility. Find the maximal and minimal elements of X.
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On a set $X\subseteq \mathbb Z^+$ you can introduce a partial order $$a \preceq b \Leftrightarrow a|b$$ (Of course you can reverse this relation, what changes the notion of swaps the notions of maximal and minimal elements).
Recall that $m\in X$ is a maximal element if there is no element greater than $m$. In this case it means that $m$ does not divide any other element of $X$. If $X$ consists of prime numbers only, this means that every element is maximal.
I leave minimal elements as an exercise.