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Math Help - Power set

  1. #1
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    Power set

    Hi, need help in proof.

    |A| =|B|, and i need to prove that |P(A)| = |P(B)|

    Thanks for advance...
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  2. #2
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    Quote Originally Posted by rebecca View Post
    Hi, need help in proof.
    |A| =|B|, and i need to prove that |P(A)| = |P(B)|
    What have you tried? You have got to show some effort.

    You know that \left| A \right| = \left| B \right|\; \Rightarrow \;\left( {\exists \alpha } \right)\left[ {\alpha :A \Leftrightarrow B} \right], a bijection.

    Is there a bijection \left[ {\beta :\mathcal{P}(A) \Leftrightarrow \mathcal{P}(B)} \right]?
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  3. #3
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    |\mathcal{P}(A)|=|2^A|, where 2^A is the set of functions from A to \{0,1\}. Proved by matching every B\subseteq A with the characteristic function of B. Now, |2^A|=2^{|A|}. Proved using the Multiplication Rule.
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