Maximum/Maximal set

Question

Maximum or maximal set with property $P$

When I was reading some textbooks, I noticed that I do not get the meaning of the following two phrases.

($P1$) $\quad$ maximum set with property $P$

($P2$) $\quad$ maximal set with property $P$

In this regard, I have the following two questions.

($Q1$) $\quad$ Are the phrases equivalent?

($Q2$) $\quad$ What are their meanings? (generally accepted meanings, meanings specific to particular theories)

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As for ($Q2$), I suspect the following three meanings.

$\quad$ A set is a $\textit{maxim... set with property}$ $P$ if and only if there is $\dots$

($M1$) $\quad \dots$ no proper superset with property $P$.

($M2$) $\quad \dots$ no set with property $P$ that has greater cardinality.

($M3$) $\quad \dots$ no other set with property $P$ that has greater or equal cardinality.

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Thanks in advance!

Answer 1

If $(A,\leq)$ is a partial order, then we define these two definitions for $a\in A$:

• $a$ is maximal if whenever $a\leq b$, then $a=b$.
• $a$ is maximum if for every $b\in A$, $b\leq a$.

You can prove that every maximum is maximal, but a maximal element need not be a maximum. In particular there can be many maximal elements. So being maximal and maximum are not the same thing in general.

Now you can consider the collection $A=\{X\mid X\text{ has property }P\}$, and $\leq$ as set inclusion. Then a maximal set with property $P$ is just a maximal element of this partial order; and a maximum is a maximum in this partial order. If those even exist, of course.