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)
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.
Thanks in advance!
$\endgroup$1 Answer
$\begingroup$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.
$\endgroup$ 8
