Finding the minimum and the maximum when speaking of relations

Question

I'm trying to figure out how to find the minimum and the maximum when speaking of relations. Consider $A=(2,3,\ldots,13)$. Also consider the following relation $R$: $$ (x,y)\in R \Leftrightarrow x|y $$ where $x,y\in A$. I need to find the minimum and the maximum of the relation. What does it mean? How do I find them?

Answer 1

$2|4$ and no element $x\in A$ other than $2$ divides $2$, thus $ 2$ is the minimum of $ A$ according to the relation $ R$.

no element of $A$ divides $ 13$ but $2|12 $ and $12$ does not divide any other element of $A$. its follows that $ 12$ is the maximum of $A$.

Answer 2

$ \newcommand{\p}{\preccurlyeq} $ A "maximal" element of a set $A$ with partial order $\p$ is an $M \in A$ such that there is no $a \in A$ where both $M \p a$ and $M \ne a$. Likewise, a minimal element of $(A,\p)$ is an $m \in a$ such that no $a \in A$ satisfies $a \p m$ and $a \ne m$ together.

Consequently, $a \in A$ is not maximal if there exists $b \in A$, $b\ne a$, such that $a \p b$. Similarly, $a$ is not minimal if $b \p a$ but $b \ne a$.

So, run through the list invalidating candidates for maxima first:

• $\not \exists x \in A$ such that $x \mid 2$ (except itself)
• $\not \exists x \in A$ such that $x \mid 3$ (except itself)
• $2 \mid 4$
• $\not \exists x \in A$ such that $x \mid 5$ (except itself)
• $3 \mid 6$
• $\not \exists x \in A$ such that $x \mid 7$ (except itself)
• $2 \mid 8$
• $3 \mid 9$
• $2 \mid 10$
• $\not \exists x \in A$ such that $x \mid 11$ (except itself)
• $2 \mid 12$
• $\not \exists x \in A$ such that $x \mid 13$ (except itself)

Thus the maxima are the primes in $A$ -- which makes sense, for the divisibility relation.

We can similarly go through the list again and eliminate candidates for minima:

• $2 \mid 4$
• $3 \mid 6$
• $4 \mid 8$
• $5 \mid 10$
• $6 \mid 12$
• $7,8,\cdots,13$ only divide themselves in $A$

Thus the maxima are $7, 8,9,10,11,12$.

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Note that there is no singular "greatest maximum"/"least minimum" for this relation. If there were, we would not have multiple maximal/minimal elements.