# 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. Mar 25, 2020 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. Mar 25, 2020 at 11:28

It comes from Bernstein's Habilitation dissertation Untersuchungen aus der Mengenlehre (1901, published 1905), where he also introduced the now common symbolism for cardinal arithmetic. The exponential notation is introduced in §2 as follows (my translation):

"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... Mar 26, 2020 at 13:02
• ... I found Hausdorff's definition extremely clear and absolutely identical to that of moderd textbook with its explicit reference to functions. So, maybe, we can say that the idea (and the notation) was actually invented by Bernstein, but the transition from (concrete) distributions to (abstract) functions was made clear in thefollowing decade and expressed in a "modern" way by Hausdorff. Mar 26, 2020 at 13:02
• @user6530 Hausdorff generally was "the first to take the sort of extensional, set-theoretic approach" (Kanamori). Although he was anticipated by Zermelo and others, extensional sets and intensional classes were routinely mixed together before Grundzüge der Mengenlehre (notice how Bernstein talks of powerclasses). I agree that Pierpont's phrasing is clearer, but you'll probably find earlier expository improvements on Bernstein in König, Zermelo, Hausdorff himself, etc., CH was popular pastime. Mar 27, 2020 at 2:11