Is continuum $ 2^{\mathbb{R}}$?

Question

I had questions if exist A $\neq \emptyset $ that |A|=|P(A)|

And my doubt is if $ |2^{\mathbb{R}}|=|\mathbb{R}|$?

Answer 1

The cardinality of the powerset $P(A)$, which is $|2^A|$, is always strictly greater than the cardinality of $A.$

$$\;|A| \lt |P(A| = \left|2^{A}\right|$$

So it follows that $|\mathbb R| \lt \left|2^{\mathbb R}\right|$.

What is true is that $|\mathbb R| = \left|2^{\mathbb Z}\right|.$