(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We're working more or less with the standard ZF axioms.

Prove that [tex]A \subseteq \mathcal{P}(\bigcup A)[/tex] for any set A, whose elements are all sets. When are they equal?

2. Relevant equations

Just the axioms

I) Extensionality

II) Emptyset and Pairset

III) Separation

IV) Powerset

V) Unionset

VI) Infinity

3. The attempt at a solution

I can prove the first part pretty easily. We have that the union and power set of the union are sets directly from V and IV.

Briefly, suppose [tex]X \in A[/tex]. Then [tex]X \subseteq \bigcup A[/tex] by def. of union. So [tex]X \in \mathcal{P}(\bigcup A)[/tex].

The part I'm stuck on is when they're equal. I attempted to prove that [tex]\mathcal{P}(\bigcup A) \subseteq A[/tex] in order to get an idea of what condition A would need to meet.

My best guess is that they're only equal when A (at the very least) contains the emptyset and the singletons of every element in the union of all of A. Seems like a pretty circular definition, though, since I need to know what A is to know what the union of all of A is. Maybe I could say that A needs to be composed of the emptyset and the remaining elements must be singletons or else be the union of singletons already contained in A? I have thought about this a fair bit, and I'm pretty sure my condition ensures equality, so this isn't a random guess. Help would be appreciated.

Also, we have not really gotten into cardinality yet, but I do know that [tex]|S| \leq |\bigcup S|[/tex] if S is countable. But I also know that [tex]|\bigcup S| < |\mathcal{P}(\bigcup S)|[/tex]. So for our equality to hold, S must be uncountable...?

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# Homework Help: Simple set theory question

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