Does $|A|^{|A|}=2^{|A|}$ hold for any infinite set $A$?

Question

Because I know the cardinality of all the functions$f:[0,1]\to \mathbb{R}$ is $2^{c}$（c is the cardinality of continuum).I wonder whether this holds generally.

Answer 1

The axiom of choice implies $\kappa^2=\kappa$ for transfinite cardinals $\kappa$. Then$$2^\kappa\le\kappa^\kappa\le(2^\kappa)^\kappa=2^{\kappa^2}=2^\kappa.$$By the Schröder–Bernstein theorem, we're done.