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2

This is not so straightforward, see Dipert, Peirce, Frege, the Logic of Relations, and Church's Theorem for a sketch of history. The notation Russell used was not created by Peano, and certainly not by Frege. Nobody used Frege's convoluted notation except his own Begriffsschrift (1879), not even Frege himself afterwards. Peano did contribute, but the basis ...


3

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 - ...


7

According to Cajori's History of Mathematical Notations, v.2, §502 (1929) the overbar, overarrow and boldface notations are older, which may be one reason why they are more common. Shaw's 1912 survey of vector notations also mentions boldface, italics, Gothics, overbar, etc., but not underbar. Judging by How to represent a tensor/matrix/vector/array in ...


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