Today the set-theoretic operations of intersection $\cap$ [German: *Durchschnitt*] and union $\cup$ [German: *Vereinigung*] are standard.


The modern notations are present in the first edition of van der Waerden's *Moderne Algebra* (1930). However, the notation is mostly missing from Steinitz' *Algebraische Theorie der Körper* (1910), which was a precursor of van der Waerden. Instead of *Menge* (for set), Steinitz uses *System*. The term *Durchschnitt* is present in Steinitz, but the word *Vereinigung* and the symbols $\cap,\cup$ are missing.


Has anything been written on the history of set-theoretic language in algebra?



Whence "Durchschnitt" and "Vereinigung"?
An extensive discussion of the origin of **"Menge"** is given in [Earliest Known Uses of Some of the Words of Mathematics](http://jeff560.tripod.com/s.html) (scroll down to "Set and Set Theory"). Cantor's (1880) [Über unendliche linear Punktmannigfaltigkeiten](https://books.google.nl/books?id=_oaABwAAQBAJ&pg=PA145) is one of the earliest uses. It contains the notation $\{\cdots\}$ for a set and introduces the term **"Vereinigung"** for the union (symbol ${\cal M}$) and **Durchschnitt** for the intersection (symbol ${\cal D}$)


![](https://ilorentz.org/beenakker/MO/cantor_2.png)


Schröder (1877) had previously used the notation $+$ and $\times$ for union and intersection. This was changed into the presently used symbols $\cup$ and $\cap$ by Peano (1888) to avoid confusion with addition and multiplication in algebra. 


[Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann](http://mathematica.sns.it/opere/138/) (1888).


![](https://ilorentz.org/beenakker/MO/peano.png)


*In this brief work by Schröder (37 pages) the mathematical logic is developed that forms the introduction of the present book. I found it useful to replace the logical symbols $\times,+,A\_1,0,1$ used by Schröder by the symbols $\cap,\cup,-A$,* ⚪, ⚫ *in order to avoid a possible confusion between symbols from logic and from mathematics (a possible confusion noted by Schröder himself). I also introduced the logical symbols $\lt$ and $\gt$, although not strictly necessary...*


