Editing Interval (mathematics) (section)

===Including or excluding endpoints===
To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described in [[International standard]] [[ISO 31-11]]. Thus, in [[set builder notation]],

:<math>\begin{align}
(a,b) = \mathopen{]}a,b\mathclose{[} &= \{x\in\R \mid a<x<b\},    \\[5mu]
[a,b) = \mathopen{[}a,b\mathclose{[} &= \{x\in\R \mid a\le x<b\}, \\[5mu]
(a,b] = \mathopen{]}a,b\mathclose{]} &= \{x\in\R \mid a<x\le b\}, \\[5mu]
[a,b] = \mathopen{[}a,b\mathclose{]} &= \{x\in\R \mid a\le x\le b\}.
\end{align}</math>

Each interval {{open-open|''a'', ''a''}}, {{closed-open|''a'', ''a''}}, and {{open-closed|''a'', ''a''}} represents the [[empty set]], whereas {{closed-closed|''a'', ''a''}} denotes the singleton set&nbsp;{{math|{''a''}{{null}}}}.  When {{math|''a'' > ''b''}}, all four notations are usually taken to represent the empty set.

Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation {{math|(''a'', ''b'')}} is often used to denote an [[tuple|ordered pair]] in set theory, the [[coordinates]] of a [[point (geometry)|point]] or [[vector (mathematics)|vector]] in [[analytic geometry]] and [[linear algebra]], or (sometimes) a [[complex number]] in [[algebra]]. That is why [[Nicolas Bourbaki|Bourbaki]] introduced the notation {{math|]''a'', ''b''[}} to denote the open interval.<ref>{{cite web|url=http://hsm.stackexchange.com/a/193|title=Why is American and French notation different for open intervals (''x'', ''y'') vs. ]''x'', ''y''[?|website=hsm.stackexchange.com|access-date=28 April 2018}}</ref> The notation {{math|[''a'', ''b'']}} too is occasionally used for ordered pairs, especially in [[computer science]].

Some authors such as Yves Tillé use {{math|]''a'', ''b''[}} to denote the complement of the interval&nbsp;{{open-open|''a'', ''b''}}; namely, the set of all real numbers that are either less than or equal to {{mvar|a}}, or greater than or equal to {{mvar|b}}.