The second line of this answer states

The elements of the poset $Z$ are families $G$ of subsets of $X$, that contain $F$ (the fixed $F$ we start out with, and which has the FIP), so we have $F \subseteq G$ for all $G \in Z$, and such that $G$ has the FIP

Question: I am confused about the fact that we have $F\subseteq G$, because $G$ is a family of subsets of $X$, which I think of as a function, while $F$ is just some subset of $X$.

That is, in few words, how can a set be a subset of a function (I realize a function is a set, but it is a set of ordered pairs. Here $F$ is not a set of ordered pairs)

Question in More Detail

Specifically, at this question ("What is the difference between a family and a set") the answer states that

Strictly speaking, a family is a function $I \to U$, where $I$ is an index set and $U$ is a universe that contains the members of the family.

Using the notation of the first block quotations, then the poset $Z$ should consist of functions $G:I\to X$, but since a function is really a collection of ordered pairs, then we can write $G=\{(i,x)\big\vert i\in I,\ x\in X\}$ (note I am slightly abusing notation in that every element of "$X$" in this definition contains $F$)

I am then unclear about how the claim $F\subseteq G$ can hold, given than $G$ is a set of ordered pairs -- albeit of which the second part contains $F$ -- while $F$ does not have ordered pairs (that is, these seem like different objects to me).


The second answer you linked is describing the term "family" as used in the article linked in its question. This is not the only possible meaning of "family"; another is that a family is just a set. That is the meaning being used in the first answer you linked: "$G$ is a family of subsets of $X$" just means "$G$ is a set of subsets of $X$".

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  • $\begingroup$ That clarifies things (although then I prefer the phrasing "G is a set of subsets of X" over "G is a family of subsets of X"). Thank you $\endgroup$ – user106860 Apr 5 '18 at 23:50

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