# Exponential notation for function spaces

If $$X$$ 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 $$Y^X \cong Y^n$$.

• 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,

• $$Z^{X \times Y} \cong (Z^X)^Y$$ (this isomorphism is called currying)

• $$Z^{X + Y} \cong Z^X \times Z^Y$$

• $$Z^1 \cong Z$$ (where $$1$$ is a one element set, since there is one function into $$Z$$ for every element of $$Z$$)

• $$Z^0 \cong 1$$ (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 $$\text{Hom}_{\mathcal{C}}(X, Y)$$ in an arbitrary cartesian closed category $$\mathcal{C}$$ for the first set of reasons listed above.

Parents:

• Mathematics

Mathematics is the study of numbers and other ideal objects that can be described by axioms.

• I don’t think this is what you mean, is it?

• Thanks, I’ve corrected it. That was a strange typo.