Cardinality of Set and its Power Set Singleton

Question

I know that I should prove that there is a bijection between the two sides but I do not know how to do it.

Answer 1

Your question is asking the followings (in the standard terminology): if $X$ is Dedekind-infinite, then $2^{2^{|X|}}=\left(2^{2^{|X|}}\right)^2$. You can prove it by using simple cardinal arithmetic: observe that $$\left(2^{2^{|X|}}\right)^2=2^{2^{|X|}\cdot 2}=2^{2^{|X|+1}}.$$ By the assumption, we have $2^{2^{|X|+1}}=2^{2^{|X|}}$.

Answer 2

HINT: Here is an outline of a proof that does not assume any knowledge of cardinal arithmetic.

Show that there is a bijection between $\wp(A)$ and $\wp(A)\times\{\wp(A)\}$. Then show that $\wp(A)$ and $\wp(A)\times\{\wp(A)\}$ are disjoint and use this fact to define a bijection between $\wp(\wp(A))\times\wp(\wp(A))$ and $\wp\big(\wp(A)\cup\big(\wp(A)\times\{\wp(A)\}\big)\big)$.

The next step is to define a bijection between

$$\wp\left(\wp(A)\cup(\wp(A)\times\{\wp(A)\}\big)\right)$$

and $\wp\big(\wp(A\cup\{A\})\big)$.

Finally, define a bijection between $\wp\big(\wp(A\cup\{A\})\big)$ and $\wp(\wp(A))$ and put the pieces together to get the desired bijection between $\wp(\wp(A))$ and $\wp(\wp(A))\times\wp(\wp(A))$.