1. ## Power Sets

Hey guys, I wanted to see if someone could check my work.

Find $\displaystyle \mathcal{P}(\mathcal{P}(\left\{1\right\}))$ and its cardinality.

$\displaystyle \mathcal{P}(\left\{1\right\})=\left\{\oslash, \left\{1 \right\}\right\}$ then,

$\displaystyle \mathcal{P}(\mathcal{P}(\left\{1\right\}))= \mathcal{P}(\left\{\oslash, \left\{1 \right\}\right\}) = \left\{\oslash, \left\{1 \right\}, \left\{\left\{1 \right\}\right\}, \left\{\oslash, \left\{1 \right\}\right\}\right\}$

$\displaystyle | \mathcal{P}(\mathcal{P}(\left\{1\right\}))| = 4$

correct?

2. ## Re: Power Sets

Originally Posted by amthomasjr
Hey guys, I wanted to see if someone could check my work.
Find $\displaystyle \mathcal{P}(\mathcal{P}(\left\{1\right\}))$ and its cardinality.
$\displaystyle \mathcal{P}(\left\{1\right\})=\left\{\oslash, \left\{1 \right\}\right\}$ then,
$\displaystyle \mathcal{P}(\mathcal{P}(\left\{1\right\}))= \mathcal{P}(\left\{\oslash, \left\{1 \right\}\right\}) = \left\{\oslash, \left\{1 \right\}, \left\{\left\{1 \right\}\right\}, \left\{\oslash, \left\{1 \right\}\right\}\right\}$
$\displaystyle | \mathcal{P}(\mathcal{P}(\left\{1\right\}))| = 4$ correct?
Yes it i is correct.

In fact if $\displaystyle A$ is a finite set then $\displaystyle \left| {\mathcal{P}(\mathcal{P}(A))} \right| = {2^{\left( {{2^{\left| A \right|}}} \right)}}$

Thank you