# Math Help - Power set

1. ## Power set

Hi, need help in proof.

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

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 $\left[ {\beta :\mathcal{P}(A) \Leftrightarrow \mathcal{P}(B)} \right]?$
3. $|\mathcal{P}(A)|=|2^A|$, 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.