Definition of the largest or smallest set that has some property

Question

Many textbooks use the term largest and smallest to describe a set that is a superset and subset, respectively, of all sets satisfying some property. The words themselves are pretty clear to deliver what they are supposed to mean, but I would like to formalize the terms to clarify the meaning. The following is my try: $ \newcommand{\imply}{\to} \newcommand{\eqv}{\leftrightarrow} \newcommand{\powset}{\mathcal{P}} \newcommand{\set}[2]{\{#1 ~|~ #2\}} $

• Largest set

Let $ P $ be a predicate that takes a subset of $ X $ and has the property: $$ \forall C \subseteq \powset(X): [\forall A \in C: P(A) \imply P(\cup C)] $$ Then, the largest subset $ L $ of $ X $ that satisfies $ P $ is a subset of $ X $ that is a superset of all sets satisfying $ P $ such that $$ L = \cup \set{B \subseteq X}{P(B)} $$ or equivalently, $ \forall x \in X: [x \in L ~\eqv~ \exists B \subseteq X: (P(B) \land x \in B)] $.

• Smallest set

Similarly, let $ Q $ be a predicate that takes a subset of $ X $ and has the property: $$ \forall C \subseteq \powset(X): [\forall A \in C: Q(A) \imply Q(\cap C)] $$ Then, the smallest subset $ S $ of $ X $ that satisfies $ Q $ is a subset of $ X $ that is a subset of all sets satisfying $ Q $ such that $$ S = \cap \set{B \subseteq X}{Q(B)} $$ or equivalently, $ \forall x \in X: [x \in S ~\eqv~ \forall B \subseteq X: (Q(B) \imply x \in B)] $.

Are these acceptable?

Answer 1

Are these acceptable?

I think the answer strongly depends on what you want to do with these concepts. You usually wouldn't formalize just for the sake of formalization - you wanna work with the definitions you come up with and when they work in that regard they're perfectly acceptable. Your idea certainly makes intuitive sense and is even used in some contexts (for example the Borel $\sigma$-Algebra over a topological space is the smallest [in the sense of inclusion - so the idea you used; although I believe your formalization is a bit off] $\sigma$-Algebra containing all open sets), but depending on the context it might also be a better idea to consider cardinality, measure or something else rather than using inclusion to order the sets.

So abstractly you could use any (pre-?)order on some class of sets (under the assumption that there is a unique maximal element statisfying your predicate wrt to that order) I believe.

EDIT: So I'd maybe phrase it like this: Let $C$ be a class of sets, $\leq$ a partial order on $C$ and $Q$ be some predicate on the elements of $C$. Furthermore assume $X := \{A \in C: Q(A)\}$ is nonempty and has a greatest element $A^*$ (see https://en.wikipedia.org/wiki/Partially_ordered_set#Extrema) with respect to $\leq$. Then we call $A^*$ the $(C, \leq)$-largest set statisfying $Q$.

I think we can define the "smallest set" via this definition by "reversing the order" (haven't thought it through though). Your definition is then the $(\mathcal{P}(X), \subseteq)$-largest (smallest) set statisfying $Q$ if I'm not mistaken.

Answer 2

This looks complicated. The terms largest/greatest and smallest/least are well defined for any poset (or even preorder).

So whenever someone talks about largest/smallest element you have to figure out what poset he is referring to. This should be deducible from context.

The typical context is when $X$ is a set and $\mathcal{F}\subseteq P(X)$ is some collection of subsets. The natural order on $P(X)$ is the inclusion, which $\mathcal{F}$ inherits. In that situation the smallest/largest element of $\mathcal{F}$ means the smallest/largest element with respect to inclusion.

Now if $Prop$ is some property and subsets of $X$ either satisfy it or not, then the smallest/largest subset of $X$ that satisfies $Prop$ is simply the smallest/largest element of

$$\mathcal{F}:=\{A\subseteq X\ |\ Prop(A)\}\subseteq P(X)$$

with respect to inclusion.

The benefit of considering abstract posets is that the terms largest/smallest can be applied to various contexts, not only to inclusions.