# Set theory question

## Homework Statement

Assume that ##x## and ##y## are members of a set ##B##. Show that ##\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B##

## The Attempt at a Solution

I know that ##\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B## iff ##\{ \{x\}, \{x,y\} \} \subseteq \mathcal{P} B##, but I don't see where this gets me. To me it's obviously true, but I don't see how to show it.

## Answers and Replies

FactChecker
Science Advisor
Gold Member
If something seems obvious, just see if you can use the basic definitions to state it. State the definition of power set and start there.
Since x and y ∈ B, {x} and {x,y} are subsets of B. By the definition of the power set ......

If something seems obvious, just see if you can use the basic definitions to state it. State the definition of power set and start there.
Since x and y ∈ B, {x} and {x,y} are subsets of B. By the definition of the power set ......
Oh, right. That seems really obvious now. So the power set of B is the set of all subsets. Since ##\{x\}## and ##\{x,y\}## are subsets of B, the set ##\{ \{x\}, \{x,y\} \}## must be a subset of the power set.