Take the tour ×  Let $\mathbf 2 = \{0, 1\}$ and $X , Y$ be sets.
(i) Describe the elements of $\mathbf 2^{X\times Y}$ and
$\left({\mathbf 2^X}\right)^Y$.  (ii) Find a bijective function
from $\mathbf 2^{X\times Y}$ to $\left({\mathbf 2^X}\right)^Y$.
(iii) Let $\mathbb N$ and $\mathbb R$ be, respectively, the sets
of natural numbers and real numbers. Describe and compare the
cardinalities of: $\mathbb N$, $\mathbb N \times \mathbb N$,
$\mathbf 2^{\mathbb N}$ , $\mathbb R$ and $\mathbf 2^{\mathbb
R}$.  First of all, apologies for being incompetent with Latex
(^) indicates "to the power of" so in the case of 2 ^ (X x Y) -
this is the set of all functions from the cartesian product of
(X x Y) into 2, 2 being defined as {0, 1}. Also; N and R are the
naturals and reals respectively.  Can someone please explain the
answers to this question? Many thanks.  share|improve this
question In your apology, you already solved half of (i). What
else did you find by yourself? –  Hagen von Eitzen Jan 12 at
14:11 Please add pa