Question

Show that C, the set of complex numbers has the same cardinality as R, the set of real numbers.

Short answer

$\left|R\right|=\left|C\right|$

Step-by-step solution

Step: 1

Let$\left|R\right|=2^{N_0}$ and$\left|C\right|=\left|RXR\right|=|R{|}^2$

Step: 2

We have$$\begin{gathered} \left|C\right|=\left|RXR\right|=|R{|}^2 \\ ={\left(2^{N_0}\right)}^2=2^{N_0-2}=2^{N_0}=|R| \end{gathered}$$

Hence $\left|R\right|=\left|C\right|$