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:
So what I want to show is:
I have the existence part down:
Suppose . Then .
Suppose and and such that .
, so clearly . Since then it
follows that , therefore . Since , and
were arbitrary, .
Where I'm having trouble is with the uniqueness proof. I believe that in order to prove uniqueness,
I have to show that
The first half of proving the equality is trivial, that is so clearly . It's the
other half that is giving me trouble, that is how do I show that ? I can see that it is true but for some reason I can't come up with well reasoned argument as to why.