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.