View source for Exponential law for sets

{{TEX|done}}{{MSC|03E20}}

The correspondence between the sets $A^{B \times C}$ and $(A^B)^C$, where $X^Y$ denotes the set of all maps from the set $Y$ to the set $X$.  Given $f \in A^{B \times C}$, that is, a [[pairing]] $f : B \times C \rightarrow A$, and given $c \in C$, let $f_c$ denote the map $f_c : B \rightarrow A$ by $f_c : b \mapsto f(b,c)$.  Then $c \mapsto f_c$ defines a map from $A^{B \times C} \rightarrow (A^B)^C$.  In the opposite direction, let $G \in (A^B)^C$.  Given $b \in B$ and $c \in C$, define $g(b,c)$ to be $G(c)$ applied to $b$.  Then $G \mapsto g$ defines a map from $(A^B)^C \rightarrow A^{B \times C}$.  These two correspondences are mutually inverse.

In computer science, this operation is known as "Currying" after Haskell Curry (1900-1982).

In category-theoretic terms, the exponential law makes the [[category of sets]] a [[Cartesian-closed category]].

In [[cardinal arithmetic]], this corresponds to the formula
$$
\mathfrak{a}^{\mathfrak{b}\mathfrak{c}} = (\mathfrak{a}^{\mathfrak{b}})^{\mathfrak{c}} \ .
$$

There are further correspondences relating the sets $A^{B \coprod C}$ and $A^B \times A^C$ and the sets $(A \times B)^C$ and $A^C \times B^C$, corresponding to the formulae in cardinal arithmetic $\mathfrak{a}^{\mathfrak{b} + \mathfrak{c}} = \mathfrak{a}^{\mathfrak{b}}\mathfrak{a}^{\mathfrak{c}}$ and $(\mathfrak{a} \mathfrak{b})^{\mathfrak{c}} = \mathfrak{a}^{\mathfrak{c}} \mathfrak{b}^{\mathfrak{c}}$.

====References====
* Benjamin C. Pierce, ''Basic Category Theory for Computer Scientists'', MIT Press (1991) {{ISBN|0262660717}}
* Paul Taylor, ''Practical Foundations of Mathematics'', Cambridge Studies in Advanced Mathematics '''59''', Cambridge University Press (1999) {{ISBN|0-521-63107-6}}