What does "Find all functions $A \rightarrow B$" mean?

Question

I keep coming across questions written in this format and I don't understand this notation. An example of a question is: Find all functions $f : \{1,2,3,\ldots\} \rightarrow \{1,2,3,\ldots\}$, such that for all $n = 1, 2, 3,\ldots$ the sum $f(1) + f(2) + ... + f(n)$ is equal to a perfect cube less than or equal to $n^3$. Another example here: How many functions $g$ can be defined from set $A = \{0, 1,\ldots, 2^n -1, 2^n\}$ to set $B = \{0, \ldots , n\}$ such that $g(2^x) = x$ for all $x \in B$? I don't need either questions solving but the notation explained.

Answer 1

It is simply specifying the domain and co-domain of the solution function. Another way of writing "find all functions $f\colon A\to B$ such that $\dots$" is "find all $f\in B^A$ such that $\dots$"

Answer 2

$f:A\to B$ means $f$ is a function from $A$ into $B$ or onto $B.$

Presumably $\{1,2,3,..\}$ means $\Bbb N,$ the set of positive integers. So the problem reads: Find all functions from $\Bbb N$ to $\Bbb N$ such that ... (etc.).

n^3 means $n^3$.

In the 2nd problem $A$ is set of non-negative integers that do not exceed $2^n$ and $B$ is the set of non-negative integers that do not exceed $n.$

Click on the "?" on the top bar (far right), choose Help Center, and click on "How Do I Format Mathematics Here?"