# Why is American and French notation different for open intervals (x, y) vs. ]x, y[?

The Americans and the French use a different notation for open intervals: The Americans use (x, y) while the French use ]x, y[. How did this notational divergence appear?

• The square brackets notation is due to Bourbaki. – Andrés E. Caicedo Oct 30 '14 at 16:56
• @AndresCaicedo Thanks, I had no idea it was so recent. Do you know why they chose this notation? – Franck Dernoncourt Oct 30 '14 at 17:42
• Some discussion on this topic here and here. One commenter suggest that the backward brackets might have been introduced by Bourbaki to prevent confusion with ordered pairs. I am still at a loss for a documented history, but it is at least a part of an old ISO standard. I am not seeing it in the latest standard ISO 80000-2. – J. W. Perry Oct 30 '14 at 18:06
• I guess it is intuitive (inclusion/exclusion of the endpoints depends on the direction of the bracket), but I have not found anything written by them stating this or some other motivation explicitly. – Andrés E. Caicedo Oct 30 '14 at 18:18
• @J.W.Perry Thanks! I had indeed went through the standard for that reason, see notation 2-6.10 last column. – Franck Dernoncourt Oct 30 '14 at 23:54

Notation $()$ is traditional, and $].[$ was introduced by Bourbaki.

Much of the Bourbaki notations and terminology became standard, but English speaking people are the most conservative ones in this respect:-) (Recall the history of the metric system:-)

Another example of the same is "injection", "surjection", "bijection". Many English authors still write "one-to-one", "onto" and "one-to-one and onto".

Another example: Bourbaki taught us that "positive" is $\geq 0$, and "strictly positive" is $>0$.

But many people still prefer "positive" to mean $>0$ and "non-negative" for $\geq0$.

Remark. I am educated in Ukraine in 1970-s, and I experienced a strong influence of Bourbaki on education. But I still like $(,)$, perhaps just for aesthetic reasons.

• Thanks, very interesting, I had no idea that the definition "positive" is $\geq 0$ was from Bourbaki as well, I am always having trouble with that in the US. – Franck Dernoncourt Nov 1 '14 at 20:14
• A friend of mine asked a member of Bourbaki, let's call him X-X. X, about this and indeed the usage is due to Bourbaki. X-X. X said Bourbaki wanted to allow the notation $\subset$ to include the possibility of equality, and not just mean a strict subset. Compatibly with that, they wanted $<$ to mean less than or equal to and $>$ to mean greater than or equal to. This is why Bourbaki started using the word positif to mean greater than or equal to 0. – KCd May 1 '15 at 4:16
• Yes, and Bourbaki had partial success: everyone uses $\subset$ nowadays in their sense. – Alexandre Eremenko May 1 '15 at 11:29
• @AlexandreEremenko: From my, admittedly rather limited, experience I'd say the opposite is (still) true: since $<$ is usually interpreted as a strict inequality, I do prefer writing $\subseteq$ for (not necessarily strict) inclusion. – Jules Lamers May 30 '17 at 14:17
• "Yes, and Bourbaki had partial success: everyone uses ⊂ nowadays in their sense." Who is everyone? If you see him, tell him that he is wrong. – Otto May 30 '17 at 20:37