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Thread: Power Sets

  1. #1
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    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?
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  2. #2
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    Re: Power Sets

    Quote Originally Posted by amthomasjr View Post
    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 $\displaystyle \left| {\mathcal{P}(\mathcal{P}(A))} \right| = {2^{\left( {{2^{\left| A \right|}}} \right)}}$
    Thanks from amthomasjr
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  3. #3
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    Re: Power Sets

    Thank you
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