# Power set

• Dec 15th 2009, 10:52 AM
rebecca
Power set
Hi, need help in proof.

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

• Dec 15th 2009, 11:22 AM
Plato
Quote:

Originally Posted by rebecca
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 $\left[ {\beta :\mathcal{P}(A) \Leftrightarrow \mathcal{P}(B)} \right]?$
• Dec 15th 2009, 11:33 AM
emakarov
$|\mathcal{P}(A)|=|2^A|$, 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.