If and $Y$ are sets, the set of functions from $X$ to $Y$ (often written $X \to Y$ ) is sometimes also written $Y^{X}$ . This latter notation, which we'll call exponential notation, is related to the notation for finite powers of sets (e.g., $Y^{3}$ for the set of triples of elements of $Y$ ) as well as the notation of exponentiation for numbers.

Without further ado, here are some reasons this is good notation.

A function $f : X \to Y$ can be thought of as an " $X$ wide" tuple of elements of $Y$ . That is, a tuple of elements of $Y$ where the positions in the tuple are given by elements of $X$ , generalizing the notation $Y^{n}$ which denotes the set of $n$ wide tuples of elements of $Y$ . Note that if $| X | = n$ , then $Y^{X} ≅ Y^{n}$ .
This notion of exponentiation together with cartesian product as multiplication and disjoint union as addition satisfy the same relations as exponentiation, multiplication, and addition of natural numbers. Namely,
- $Z^{X \times Y} ≅ (Z^{X})^{Y}$ (this isomorphism is called currying)
- $Z^{X + Y} ≅ Z^{X} \times Z^{Y}$
- $Z^{1} ≅ Z$ (where $1$ is a one element set, since there is one function into $Z$ for every element of $Z$ )
- $Z^{0} ≅ 1$ (where $0$ is the empty set, since there is one function from the empty set to any set)

More generally, $Y^{X}$ is good notation for the exponential object representing ${Hom}_{C} (X, Y)$ in an arbitrary cartesian closed category $C$ for the first set of reasons listed above.

Exponential notation for function spaces