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What is the cardinality of all inverse functions defined on: $\mathbb{R}\rightarrow \mathbb{R}$?

easy to calculate the upper bound which is $2^\aleph$. ($\aleph$ is the cardinality of the continuum)

I need to find a lower bound. Probably by finding an injective function and deduce by Cantor–Bernstein–Schroeder theorem that the cardinality is $2^\aleph$.

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Hint: There are $\aleph$ positive reals. For any subset $A$ of the positive reals, define a bijection $\phi_A$ from $\mathbb{R}$ to $\mathbb{R}$ as follows. Let $\phi_A(0)=0$. If $a\in A$, let $\phi_A(a)=a$ and $\phi_A(-a)=-a$. If $a\not\in A$, let $\phi_A(a)=-a$ and $\phi_A(-a)=a$.

This gives a bijection between the set of subsets of the positive reals and a subset of the set of bijections.

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  • $\begingroup$ Got it! how can one come up with this idea? $\endgroup$ – AnnieOK Jun 17 '14 at 9:01
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    $\begingroup$ There are many others that would work. I was trying to get a sort of "indicator" (yes/no) function. Don't flip if yes, flip if no sounded like a good idea. $\endgroup$ – André Nicolas Jun 17 '14 at 9:04

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