I need help proving that if ${A}\sim{B}$ then ${P(A)}\sim{P(B)}$
Proof of equivalence between power sets
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0@Asaf thank you for your advice. Personally I do believe in what you say. – 2012-04-11
1 Answers
Let $f:A\to B$ be a bijection. Define $F:\wp(A)\to\wp(B):S\mapsto\{f(s):s\in S\}$. Prove that $F$ is a bijection.
Added: It may be useful to suggest how you might come up with this idea. You could look at some small examples. For instance, $A=\{1,2,3\}\sim\{a,b,c\}=B$, and you could write down a bijection from $A$ to $B$, say
$\begin{align*} &1\mapsto a\\ &2\mapsto b\\ &3\mapsto c\;. \end{align*}$
Now look at a ‘typical’ member of $\wp(A)$, perhaps $\{1,3\}$. You want to match it up with a subset of $B$. It seems to me that there’s just one natural candidate, namely $\{a,c\}$:
$\begin{align*} &\color{red}{1\mapsto a}\\ &2\mapsto b\\ &\color{red}{3\mapsto c}\;. \end{align*}$
Explore this idea just a little, and you should quickly see that it works in general.