- #1

omoplata

- 327

- 2

## Homework Statement

Prove that [itex]\cup_{x \in C} \{ 2^{x} \} \subseteq 2^{\cup C}[/itex]

## Homework Equations

[itex]\cup_{x \in C} \{ 2^{x} \} = \{ A | \exists x \in C, A \subseteq 2^{x} \}[/itex]

[itex]2^{x}[/itex] is the powerset of [itex]x[/itex]. i.e. [itex]2^{x} = \{ y | y \subseteq x \}[/itex]

## The Attempt at a Solution

Suppose [itex]A \in \cup_{x \in C} \{ 2^{x} \}[/itex]. Then,

[itex]\exists x \in C, A \in 2^{x}[/itex]

[itex]\exists x \in C, A \subseteq x[/itex]

[itex]A \subseteq ( \cup C )[/itex]

[itex]A \in 2^{\cup C}[/itex]

Therefore, [itex]A \in \cup_{x \in C} \{ 2^{x} \} \Rightarrow A \in 2^{\cup C}[/itex]

Therefore, [itex]\cup_{x \in C} \{ 2^{x} \} \subseteq 2^{\cup C}[/itex]

But I think there might be something wrong with my proof. Because why can't I start assuming [itex]A \in 2^{\cup C}[/itex] and go to [itex]A \in \cup_{x \in C} \{ 2^{x} \}[/itex]. That means [itex]A \in 2^{\cup C} \Rightarrow A \in \cup_{x \in C} \{ 2^{x} \}[/itex] and therefore [itex]2^{\cup C} \subseteq \cup_{x \in C} \{ 2^{x} \}[/itex] also, which means [itex]\cup_{x \in C} \{ 2^{x} \} = 2^{\cup C}[/itex].

Is there something wrong with this proof?