Hi, need help in proof.
|A| =|B|, and i need to prove that |P(A)| = |P(B)|
Thanks for advance...
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 $\displaystyle \left[ {\beta :\mathcal{P}(A) \Leftrightarrow \mathcal{P}(B)} \right]?$
$\displaystyle |\mathcal{P}(A)|=|2^A|$, 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.