# What does power-set mean here?

Let $\mu (V) = V_1 \cup V_2 \cup \cdots \cup V_k$, where each $|V_i| \le n$; means each $V_i$ contains at most $n$ points and Intersection between any two $V_i$ and $V_j$ is empty.

What does $P(\mu (V))$ mean? Does it mean $P(V_1,V_2, \ldots, V_k)$ or anything else? Is it true that a typical element in power-set will be $\{V_1,V_2\}$. Are these two valid elements in a power -set? • No. Elements of the power set will contain members of the $V_n$'s, not the $V_n$'s themselves. – Malice Vidrine Sep 17 '17 at 17:00
• I wonder if you saw $\bigcup V$ and copied it as $\mu V. \qquad$ – Michael Hardy Sep 17 '17 at 17:22
• @ Michael Hardy No – user437890 Sep 17 '17 at 17:51

By what you wrote, $\mu(V)$ consists of all elements which appear in one of the $V_i$'s. Consequently, elements of the powerset $P(\mu(V))$ are just sets $U$ such that all elements of $U$ are elements of $V_i$'s. In particular, $\{V_1,V_2\}$ is (usually) not be an element of the powerset, but for example $V_1\cup V_2$ is.
EDIT: Since you added a new question with a picture in your original post: Yes, these two sets drawn there are elements of the powerset, since they contain only elements which are also elements of the $V_i$'s. The second set drawn there is $V_4\cup V_5$, which is close to the example I gave.
• This is allowed, for example the set 1 in your picture. It has to by any collection of elements of the $V_i$'s. – x432ph Sep 17 '17 at 17:14