# Thread: Trouble with unique existance proof

1. ## Trouble with unique existance proof

Exercise 3.7.1 of "How To Prove It" (Vellemen 2006)

Suppose F is a family of sets. Prove that there is a unique set A that has the following two properties:

(a) $\displaystyle F \subseteq \mathcal{P}(A)$
(b) $\displaystyle \forall B (F \subseteq \mathcal{P}(B) \rightarrow A \subseteq B)$

So what I want to show is:

$\displaystyle \exists !A[F \subseteq \mathcal{P}(A) \land \forall B(F \subseteq \mathcal{P}(B) \rightarrow A \subseteq B)]$

I have the existence part down:

Suppose $\displaystyle A = \cup F$. Then $\displaystyle F \subseteq \mathcal{P}(A)$.

Suppose $\displaystyle y \in F$ and $\displaystyle x \in y$ and $\displaystyle B$ such that $\displaystyle F \subseteq B$.

$\displaystyle A = \cup F$, so clearly $\displaystyle x \in A$. Since $\displaystyle F \subseteq \mathcal{P}(B)$ then it
follows that $\displaystyle x \in B$, therefore $\displaystyle A \subseteq B$. Since $\displaystyle x$, $\displaystyle y$ and
$\displaystyle B$ were arbitrary, $\displaystyle \forall B(F \subseteq \mathcal{P}(B) \rightarrow A \subseteq B)$.

Where I'm having trouble is with the uniqueness proof. I believe that in order to prove uniqueness,
I have to show that

$\displaystyle \forall C \{ [F \subseteq \mathcal{P}(C) \land \forall B(F \subseteq \mathcal{P}(B) \rightarrow C \subseteq B)] \rightarrow C = \cup F \}$

The first half of proving the equality is trivial, that is $\displaystyle F \subseteq \mathcal{P}(C)$ so clearly $\displaystyle \cup F \subseteq C$. It's the
other half that is giving me trouble, that is how do I show that $\displaystyle C \subseteq \cup F$? I can see that it is true but for some reason I can't come up with well reasoned argument as to why.

2. I'm tired and completely didn't think about $\displaystyle A = \cup F$. I also haven't done set theory in awhile now. Good insight! You are right. The first half of the uniqueness proof is trivial since it follows from (b) and that A exists with the given properties. Given the definition you provided for uniqueness, wouldn't A be included in the $\displaystyle \forall B$? If so, does it now follow from (a) then that:

$\displaystyle F \subseteq P(A) \Rightarrow C\subseteq A$

Then the proof is complete, assuming everything else is correct.

3. Originally Posted by bryangoodrich
I'm tired and completely didn't think about $\displaystyle A = \cup F$. I also haven't done set theory in awhile now. Good insight! You are right. The first half of the uniqueness proof is trivial since it follows from (b) and that A exists with the given properties. Given the definition you provided for uniqueness, wouldn't A be included in the $\displaystyle \forall B$? If so, does it now follow from (a) then that:

$\displaystyle F \subseteq P(A) \Rightarrow C\subseteq A$

Then the proof is complete, assuming everything else is correct.
Hey that's great Yeah, that completes the proof. I don't know why, but sometimes if I don't get how to do a certain proof right away, I'll get stuck in this awful whirlpool of circular reasoning that I just can't seem to get out of.