What is the set of all functions from ℤ to ℤ?

Recall that a function between two sets (say $A,B$) is simply a binary relation that associates to every element of the first set $A$ exactly one element of the second set $B$. Therefore, a function can be represented as a set $G$ of ordered pairs $(x,y), x in A, y in B$, satisfying the following: each element of $A$ appears exactly once in this set (to make sure there are no repetitions and that a single element cannot be mapped to two different outputs, i.e., this map is a function).

Now, to count the total number of functions between two sets we need to count the number of unique sets (of the form $G$ above) that can be constructed. One can show that the number of functions from a set $A$ to a set $B$ is $|B|^{|A|}$ (see, for example, this intuitive answer).

Do the same for $A = mathbb{Z} = B$ but beware of their cardinalities (and realize that this set is uncountable, see for example, here).

Update (from the comments): The notation ``set of all functions from $mathbb{Z}$ to $mathbb{Z}$'' is shorthand for ${f | f: mathbb{Z} rightarrow mathbb{Z}}$, i.e., all functions whose domain and range is $mathbb{Z}$ (or subsets of $mathbb{Z}$).