Subset partial order

What is subset partial order ? Now in my opinion one thing where John Velleman’s fails to explain properly is this. Let’s start from the definition of smallest element to understand it:

Suppose R is a partial order on a set A, B ⊆ A, and b ∈ B. Then b is called an R-smallest element of B if ∀x ∈ B(bRx).

Ok, that’s cool. And now there is this family of set on which it is asked to find the smallest set: F = {X ⊆ R ∣ 5 ∈ X ∧ ∀x∀y((x∈X∧x<y)⇒y∈X)}.

Ok, so basically F is a set whose elements are subset of R with some property. Now here’s where Velleman strikes with his explanation: “smallest means smallest with respect to the subset partial order”.

The answer seems much simpler. Subset partial order means partial order in which ⊆ is the relation. So in the above case the smallest in F means an element X which satifises this property:

∀Y ∈ F(X⊆Y)

Now, I think what confused me was the initial statement: “when comparing subsets of some set A, then mathematicians use the parial order S = {(X,Y) ∈ P(A) × P(A) ∣ X ⊆ Y}”. And this made me always figure out the set S for any statement related to smallest element. So is there any set S for F ?

Now F is a set which contains element which itself is a set. So S must be definitely a relation on P(R). So the definition of S goes like this:

S = {(X,Y) ∈ P(R) × P(R) ∣ X ⊆ Y}

The key thing to understand here is that F ⊆ P(R).