Subset partial order

April 16, 2015

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 ∧ ∀xy((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(AP(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(RP(R)∣X ⊆ Y}

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