Let there be 2 sets $X$ and $Y$, both are countable (assume the bijection from $\mathbb{N}$ to the respective sets is computable) and infinite. Let $S$ be the set of all possible functions (NOT necessarily computable, i dont care about computability at this point) from $A$ to $Y$ for all $A \subset X$ (Remember that $A$ is not fixed here and it varies to cover all subsets of $X$. Note that $X$ and $Y$ are fixed in this question).I understand that $S$ is uncountable (as the number of subsets of $X$ will be uncountable) and hence all functions in it cannot be computable because there are countable number of computable functions. My question is that if there are indeed uncomputable functions in $S$, can somebody please give me an example of one such function or example of a subset of $S$ which contains only uncomputable functions ?
Thankyou!