Showing when a subset is an equality

[Mr Davis 97]

Homework Statement

In general, $A \subseteq \mathcal{P} \bigcup A$. Under what conditions does equality hold?

Homework Equations

The Attempt at a Solution

I can't seem to figure this out. If $A$ has $n$ elements, then clearly $| \bigcup A | \ge n$, which would mean that $| \mathcal{P} \bigcup A | \ge 2^n$, right? In this case the cardinalities never seem to be the same, in which case equality can never hold.

 

[mfb]

$\mathcal{P} \bigcup A$ and $\bigcup \mathcal{P} A$ are different things.

 

[Mr Davis 97]

$\mathcal{P} \bigcup A$ and $\bigcup \mathcal{P} A$ are different things.

Oops. In both cases I meany $\mathcal{P} \bigcup A$

 

[mfb]

What is $\bigcup A$? The union of what, if A is a single object?