# $\bigcup_{i \in I} \mathcal{P} (A_i)$

This is Velleman 3.7, Problem 4

Below is the problem, verbatim.

Suppose $\{ A_i \mid i \in I\}$ is a family of sets. Prove that if $\mathcal{P}(\bigcup_{i \in I} A_i) \subseteq \bigcup_{i \in I} \mathcal{P}(A_i)$, then there is some $i \in I$ such that $\forall j \in I (A_j \subseteq A_i)$.

My question is not on the proof itself, but on the notation.

If a concrete instance of the family were $\{ \{2\}, \{2, 3\}\}$, are the following correct?

$\bigcup A_i = \{2, 3\}$

$\mathcal{P}(\bigcup_{i \in I} A_i) = \{ \emptyset, \{2\}, \{3\}, \{2, 3\}\}$

$\mathcal{P}(A_i) = \{ \emptyset, \{\{2\}\}, \{\{2, 3\}\}, \{ \{2\}, \{2, 3\} \} \}$

$\bigcup_{i \in I} \mathcal{P}(A_i) = \{ \{2\}, \{2, 3\} \}$

I want to make sure I understand the problem first before attempting the proof.

• what is $A_i$? ? Commented Jan 18, 2015 at 15:37
• The problem statement only said that its a family of sets. Commented Jan 18, 2015 at 15:42
• The question is not clear. What is the relation between $A$ and $A_i$? Why did you choose such $A$ and what is the meaning of "then I believe $\cup A_i = \cdots$"? Please update the question. Commented Jan 18, 2015 at 16:00
• @Krish I updated the question. Hope it's clearer now. Commented Jan 19, 2015 at 0:05
• a possible duplicate of math.stackexchange.com/questions/1093822/… You can find an answer in this link. I hope it will help you. Commented Jan 19, 2015 at 7:03

I got the answer from Mathematica.

Clear[A, B, F];

A = Sort[{{2}, {2, 3}}];

Union[A] = {{2}, {2, 3}}

Subsets[Union[A]] = {{}, {{2}}, {{2, 3}}, {{2}, {2, 3}}}

Subsets[A] = {{}, {{2}}, {{2, 3}}, {{2}, {2, 3}}}

Union[Subsets[A]] = {{}, {{2}}, {{2, 3}}, {{2}, {2, 3}}}