Let $g: P \mapsto Q$ and $h: R \mapsto S$ be bijective.

Question

Let $g: P \mapsto Q$ and $h: R \mapsto S$ be bijective functions. Give a bijection $X: R^P \mapsto S^Q$ where $M^N$ means the set of all functions from $N$ to $M$.

I think I am missing something. I know since $X$ is supposed to be a bijection, the size of the set of all functions from $P$ to $R$ is the same as the set of all functions from $Q$ to $S$. Is it possible to label them and create a bijection using the Enumeration Principle? Am I correct in thinking this? Could someone provide such a particular bijection?

Answer 1

There is a natural map between $f:R^{P}\rightarrow S^{Q}$ given by $$f(\varphi)=h\circ\varphi\circ g^{-1}\qquad \varphi\in R^{P}$$ which has inverse $$f^{-1}(\varphi)=h^{-1}\circ\varphi\circ g\qquad \varphi\in S^{Q},$$ hence $f$ is a bijection.

Answer 2

Define a function $\phi:R^P\to S^Q$ by the prescription: $$u\mapsto h\circ u\circ g^{-1}$$

Define a function $\psi:S^Q\to R^P$ by the prescription: $$v\mapsto h^{-1}\circ v\circ g$$

Then it is not difficult to prove that $\phi\circ\psi$ and $\psi\circ\phi$ are both indentities.

From this we conclude that $\phi$ and $\psi$ must be bijections.