# History of exponential notation for the set of functions between two sets

It's well-known that if $$A$$ and $$B$$ are two sets, then the set of all functions from $$A$$ to $$B$$ can be denoted by $$B^A$$: explanations of this particular notation can be found in many places:

What I ask for is: when this notation was first introduced and in which context? (So this question is not about the meaning or the rationale behind it.)

The older occurrence I can find is in Bourbaki's Théorie des ensembles, of 1954, E.R.20, but is it the first?

• I believe the notation was widely used by Polish mathematicians in the 1920s and 1930s (Kuratowski, Sierpinski, etc.), and probably a scan of early volumes in Fundamenta Mathematica will give many pre-1954 uses (even pre-Bourbaki uses). However, all those are are preceded by the usage in Hausdorff's famous 1914 book on set theory and topology (see pp. 37 and 40-41). I don't know if Hausdorff was the first to use this notation. – Dave L Renfro Mar 25 at 11:10
• Slightly earlier appearance: pp. 295-299 (especially p. 295, Article 255, #2) in The Theory of Functions of Real Variables. Volume II by James Pierpont (1912). I suggest looking through Cantor's papers (link given in this answer), as it's starting to appear that the notation might go all the way back to him. – Dave L Renfro Mar 25 at 11:28

"If $$M$$ and $$N$$ are two sets, we call that set which - in the sense of a known expression - contains all combinations of elements from $$M$$ to the classes of $$N$$, the power $$M^N$$ ($$M$$ raised to $$N$$). Regarding the application to addition, multiplication and powerclasses of commutative and associative laws, they are the same as for finite numbers".
Bernstein does not use $$2^N$$ for the powerset, but he does write $$2^{\aleph_\alpha}$$, meaning set cardinality, in §9, when discussing the continuum hypothesis.
• Thanks a lot. If I correctly understand, Pierpont gave an expandend (and clearer) version of Bernstein's definition, which actually refers to combinations or distributions, rather than functions. Indeed, for Bernstein $M$ ($K$ for Pierpont) is not simply a set, but a set of (equivalence) classes. If $M$ and $N$ are finite, it's quite obvious to translate a single distibution into a function form $N$ to $M$, but it is less obvious what a distribution between infinite sets is (and maybe that's why Bernstein do not use $2^\mathbb{N}$). On the other hand... – user6530 Mar 26 at 13:02