Let $A$ and $B$ two infinite sets and let $|A| = m$ and $|B| = n$ and also suppose that $m > n$. Also suppose that $g$ is an injective function from $B$ into $A$. For each of the following sets, find its cardinality $(m, n, 2^m \mbox{ or } 2^n)$ and prove your answer.
1) $S_{1} = \{f\in B^A : |f(A)| = 1 \}$.
2) $S_{2} = \{ f\in B^A : |f(A)| = 2 \}$.
3) $S_{3} = A - g(B) $
I am trying to solve this exercise but i seem to struggle a lot on proving the cardinality of the above sets.
Hint: The first one asks how many functions from $A$ to $B$ there are where every element of $A$ is mapped to the same element of $B$. That is what $|f(A)|=1$ means. How many functions like that are there? The second asks how many functions from $A$ to $B$ there are where every element of $A$ is mapped to one of two elements of $B$. How is this different from $S_1?$ What do you think $S_3$ means?