# Bourbaki and the symbol for set inclusion

Which notation ($\subset$ or $\subseteq$) was preferred by Bourbaki for set inclusion (not proper)?

A side question: Was the notation for subset one of the many notations invented by Bourbaki?

• And why exactly did someone vote to close this question? – Gaussler Apr 27 '15 at 8:56
• Someone thinks this is primarily opinion based; which I do not agree, as I think the answer is "well-defined." – awllower Apr 27 '15 at 8:57
• Yes, indeed, I'd say there is a definite, opinion-free answer to this. Whether it should have been on this SE or "History of Science and Mathematics" is another matter. – Gaussler Apr 27 '15 at 8:58
• Someone possibly overread "Bourbaki" or didn't care. I've added the math-history tag to convey the intention in the tags as well. I suggest clarifying or removing the second question, though. That one is opinion-based. – AlexR Apr 27 '15 at 8:58
• Those votes to close make absolutely no sense. If someone thinks this is a bad question, they should simply downvote. – goblin Apr 28 '15 at 10:16

See :

Définition 1 (L'inclusion). La relation désignée par $(\forall z)((z \in x) \implies (z \in y))$ dans laquelle ne figurent que les lettres $x$ et $y$, se note de l'une quelconque des manières suivantes : $x \subset y, y \supset x$, « $x$ est contenu dans $y$ », « $y$ contient $x$ », « $x$ est un sous-ensemble de $y$ ».

See English translation :

Regarding origins :

According to Florian Cajori (A History of Mathematical Notations (1928), vol. 2, page 294), the symbols for "is included in" (untergeordnet) and for "includes" (übergeordnet) were introduced by Ernst Schröder : Vorlesungen über die Algebra der Logik. vol. 1 (1890).

In addition, Schröder uses $=$ superposed to $\subset$ for untergeordnet oder gleich, i.e. $\subseteq$; see Vorlesungen.

Giuseppe Peano, in Arithmetices Principia Novo Methodo Exposita (1889), page xi, uses an "inverted C" for inclusion :

Signum $\text {"inverted C"}$ significat continetur. Ita $a \ \text {"inverted C"} \ b$ significat classis $a$ continetur in classis $b$ [i.e. : $\forall x(x \in a \to x \in b)$].

Note. It is worth noticing that in Peano there is the distinction between the relation : "to be an element of" ($\in$) and the relaion : "to be included into" ($\subset$).

This distinction is not present in Schröder.

According to Bernard Linsky, Russell’s Notes on Frege’s Grundgesetze Der Arithmetik from §53, in Russell, 26 (2006), page 127–66 :

Gregory Moore reports that Russell used $\supset$ for class inclusion as well as implication until March or April 1902, when he started to use $\subset$ for class inclusion.

• And did Schröder write $\subset$ or $\subseteq$ for non-proper inclusion? – Gaussler Apr 27 '15 at 10:51