I got this question in homework:
Let $\{0,1\}^A$ the set of all functions from A (not necessarily a finite set)
to $\{0,1\}$. Find a correspondence (function) between $\{0,1\}^A$ and
$\mathcal P(A)$ (The power set of $A$).
Prove that this correspondence is one-to-one and onto.
I don't know where to start, so I need a hint. What does it mean to find a correspondence?
I'm not really supposed to define a function, right?
I guess once I have the correspondence defined somehow, the proof will be easier.
Any ideas? Thanks!
Best Answer
This is essentially the same as Martin and yuone's answers:
Fix a set $A$. For a function $f$ from $A$ to $\{0,1\}$, let $ A_f$ be the set of elements of $A$ that are mapped to 1 by $f$. That is, $a\in A_f$ if and only if $f(a)=1$.
Consider the map $\Phi(f) =A_f$.
Now if $f\ne g$, there is an $a\in A$ with $f(a)=0$ and $g(a)=1$ (or $f(a)=1$ and $g(a)=0$).
Then $A_f\ne A_g$. So $\Phi$ is one-to-one.
Now let $B\in{\cal P}(A)$. Define $f(x)=\cases{1,&x\in B\cr 0,&x\notin B }$
Then $\Phi(f)=B$. This shows that $\Phi$ is onto ${\cal P}(A)$