Hi, need help in proof.

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

Thanks for advance...

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- Dec 15th 2009, 09:52 AMrebeccaPower set
Hi, need help in proof.

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

Thanks for advance... - Dec 15th 2009, 10:22 AMPlato
What have you tried? You have got to show some effort.

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

Is there a bijection $\displaystyle \left[ {\beta :\mathcal{P}(A) \Leftrightarrow \mathcal{P}(B)} \right]?$ - Dec 15th 2009, 10:33 AMemakarov
$\displaystyle |\mathcal{P}(A)|=|2^A|$, where $\displaystyle 2^A$ is the set of functions from $\displaystyle A$ to $\displaystyle \{0,1\}$. Proved by matching every $\displaystyle B\subseteq A$ with the characteristic function of $\displaystyle B$. Now, $\displaystyle |2^A|=2^{|A|}$. Proved using the Multiplication Rule.