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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).

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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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