Collections (families of sets) and powersets I've recently been able to prove that $P(E) \cap P(F) = P(E \cap F)$ and a few similar dealing with powersets. However the notation of collections is rather confusing, and I'd like to have a clear understanding.
Ive been asked to show E is always equal to $E = \bigcup P(E)$. At best I believe I can interpret this as $\{X: X \in P(E)\ or\ X \in X\}$. Note that the question may also be written E = $\bigcup_{X \in P(E)} X$. Which makes even less sense. Could someone go into detail into the notation of collections of intersections/unions and its use with this question?
 A: The notation $\bigcup P(E)$ does not mean what you said.  
In general, $$\def\S{\mathscr S}\bigcup_{S\in \S} S$$ means the union of all the sets in $\S$, which is a family of sets. The notation $$\bigcup \S$$ (with no subscript) is a shorthand for this. Some entity $x$ is an element of  this union if it is an element of one of the sets in the union. 
This is the way unions always work.  For example, $x$ is an element of the union $A\cup B\cup C$ if it is an element of one of the sets in the union; that is if it is a element of $A$ or an element of $B$ or an element of $C$. 
So:  $$x\in \bigcup_{S\in\S} S\\\text{means}\\ x \in S \text{ for some $S\in \S$.}$$
Or using the shorthand:
$$x\in \bigcup \color{darkblue}{\S}\\\text{means}\\ x \in S \text{ for some $S\in \S$.}$$
(That was the crucial point, which you should make sure you understand before continuing.)
Applying this when $\S = P(E)$ we have $$x\in \bigcup \color{darkblue}{P(E)}\\\text{means}\\ x \in S \text{ for some $S\in \color{darkblue}{P(E)}$.}$$
But $S\in P(E)$ is true exactly when $S\subset E$; this is the definition of $P(E)$.  So we can rewrite the formula above as:
$$x\in \bigcup P(E)\\\text{means}\\ x \in S \text{ for some $\color{darkblue}{S\subset E}$.}$$
If at this point you could show that $$x\in S\text{ for some } S\subset E$$ and $$x\in E$$ were equivalent conditions, you would be done, because then you would have
$$x\in \bigcup P(E)\\\text{is equivalent to}\\ x \in E.$$
which is what you were asked to show.
I hope this is what you were looking for.
