Let $S$ be the set of all possible functions mapping $\{\sqrt 2, \sqrt 3, \sqrt 5, \sqrt 7 \}$ to $\Bbb Q$, find the cardinality of $S$.
At first I wanted to use the theorem that for any sets $A$ and $B$, the cardinality of the set of all functions mapping $A$ to $B$ is $\vert B \vert ^ {\vert A \vert}$, but in the proof of this theorem when $A$ and/or $B$ is infinite, words like "$\aleph_0$ possibilities of each elements in $A$ mapping to $B$" are used. While my instructor thought we can’t definitely say which cardinal number $\vert B \vert ^ {\vert A \vert}$ actually is.
So without using the above theorem, is there any other way to prove the question? i.e from proving bijection of $S$ to $\Bbb Q$?
Best Answer
It's clearly in bijective correspondence with $\Bbb Q^4$: if $f$ is such a function, map it to $(f(\sqrt{2}),f(\sqrt{3}),f(\sqrt{5}),f(\sqrt{7})) \in \Bbb Q^4$. The function $f$ is completely determined by these 4 rational values, and all tuples define such a function.
The square $A^2$ of a countable set $A$ is countable, and we can apply this twice to the countable set $\Bbb Q$ to get that $S\simeq \Bbb Q^4$ is countable (and infinite, so of size $\aleph_0$).