Show that all codings can be transformed into codings with alphabet $\Sigma=\{a\}$

Question

I have trouble understanding this exercise, especially since we never defined what it means for a coding to be 'transformed' into another one.

It seems to me as if the question is just: Can you count just by listing ones (or $a$'s)? In this case the answer is yes, I assume. Or to put it slightly differently: Is there a bijection between all words over some finite alphabet $\Sigma$ and all words over $\{a\}$? Again, I think of course there is, as by listing all words in $\Sigma$ one after another one is essentially counting them up, so they correspond to the list $\{a, aa, aaa, ..\}$.

Do you think this is the right direction or am I completely misunderstanding the whole point? If the former, what is really meant by 'showing' this, I mean what do they expect to read, there isn't really much one can say about it, or so it seems to me...

Thanks!

Answer 1

It depends how exactly they define coding and what the exact context of this exercise is. But from the limited context which you present here, your line of thought seems correct to me. I cannot be sure though.