# Subset partial order

April 16, 2015What 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*(*b**R**x*).

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*).