0

I am studying for an exam in elementary set theory and I am not understanding the proof for this theorem:

For any three sets $A,B,C$:

$(A^B)^C \sim A^{(B \times C)}$

I know I need to find a bijective function $G : (C\rightarrow A^B) \rightarrow (B\times C \rightarrow A)$. But I don't have a clue.

Any help would be appreciated. Thanks

ikoikoia
  • 329

1 Answers1

0

Let $f\colon C\to A^B$. Define $\varphi(f)\colon B\times C\to A$ to be the function $$ \varphi(f)\colon (b,c)\mapsto f(c)(b) $$ (the value of $f(c)$ at $b$), which is a function $\varphi(f)\colon B\times C\to A$.

Let $g\colon B\times C\to A$; for $c\in C$, define $\psi(g)\colon C\to A^B$, by declaring that, for $c\in C$, $$ \psi(g)(c)\colon b\mapsto g(b,c) $$ so we have a function $\psi(g)\colon C\to A^B$.

egreg
  • 238,574