Partitioning an infinite set into fixed number of sets

Question

Suppose we have a set of size $\kappa$, and want to partition it into $\mu$ sets, where $\kappa$ is an infinite cardinal, and $1<\mu\leq\kappa$. I am aware that it can be done in $2^\kappa$ ways (in the standard ZFC theory), but I cannot find a reference for this, nor a simple explanation.

Answer 1

There are $2^\kappa$ ways of dividing $\kappa$ into $2$ sets. To see why, note that $\{A,\kappa\setminus A\}$ is a partition of $\kappa$, and the function mapping $A\mapsto\{A,\kappa\setminus A\}$ is a $2$-to-$1$ function. By standard cardinal arithmetic, this means that the image of the function is of size $2^\kappa$.

On the other hand, if we have a partition of $\kappa$, then we can represent it as a function $f\colon\kappa\to\kappa$. In fact, as many functions of that form. But there are only $\kappa^\kappa=2^\kappa$ such functions, which gives us the upper bound of $2^\kappa$. Therefore, there are exactly $2^\kappa$ partitions, even to $\kappa$ parts.

So in particular, partitioning $\kappa$ into $\mu$ parts has $2^\kappa$ to go about that.