Let $X$ be a set. $X^X$ denotes $\{f:X\rightarrow X\}$.
I know how to show that $|\mathbb N^\mathbb N|=|2^\mathbb N|$ and that $|\mathbb R^\mathbb R|=|2^\mathbb R|$.
Hypothesis: Let $X$ be an infinite set. Then $|X^X|=|2^X|$.
In order to prove this we shall try Cantor-Bernstein's theorem. It's evident that $|2^X|\leqslant|X^X|$ (we can embed $2\hookrightarrow X$ and then $2^X\hookrightarrow X^X$). But I have no idea how to build $X^X\hookrightarrow 2^X$. I cannot generalize the method I used for $\mathbb N^\mathbb N$ and $\mathbb R^\mathbb R$ because it significantly depends on representations of natural and real numbers.
I don't even know wether it is true or not or, for instance, if it depends of continuum hypothesis.
