${\,}| {\Bbb{R}} {\rightarrow} {\Bbb{N}} {\,}| = {\,}| {\Bbb{N}}^{\Bbb{R}} {\,}| = \aleph_0^{\aleph} = \aleph_0{^2}^{\aleph_0}$
${\,} | {\Bbb{R}} {\times} ({\Bbb{R}} \rightarrow \{0,1\}) {\,}| = |{\Bbb{R}}| \cdot|{\Bbb{R}}\rightarrow\{0,1\}| = \aleph \cdot 2^{\aleph} = 2^{\aleph_0} \cdot 2^{\aleph} = 2^{\aleph_0 + \aleph} = 2^{\aleph}$
$\aleph_0^{\aleph_0} = \aleph \, \,$ ?
Any more ways to simplify the expressions ? are they right ?
Any references for more complex cardinal arithmetic ?
Thank you.