Cardinality of the set of all functions with finite support

Question

Let F be a countable field and B an infinite set. Let $(F^B)_0$ be the set of all functions with finite support from F to B. Is it true that $|(F^B)_0|=|B|$?

Answer 1

Assuming the axiom of choice, yes.

To see this note that there is an injection from $B$ into $(F^B)_0$ in the obvious way; in the other direction $(F^B)_0$ is a subset (or rather, can be identified with a subset) of $(B\times F)^{<\omega}$, the set of all finite sequences from $B\times F$.

The cardinality of $(B\times F)^{<\omega}$ is $\sum_{n\in\Bbb N}|B\times F|=|B|\cdot|F|\cdot\aleph_0$, but since $B$ is infinite, you get that this just equals $|B|$.