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I read on Mathworld they are called half-closed interval; however, it doesn't tell me how to say it in English. Also, how does one denote which one of the two options that is the one the one refering too. (I don't know how to say that better).

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Honestly, I don't think there is a very good way to express this with just spoken words only. These things are called "half-open" or "half-closed," and to get the fine points across you need to write the notation, or say something like "excluding a, including b" or make other sorts of hand motions to get the point across. – rschwieb Sep 6 '13 at 15:10
I like to call the latter $a$ unto $b$ at least in my mind. Unfortunately, this will generally not be understood and it sounds just too biblical... – leftaroundabout Sep 7 '13 at 2:27
up vote 19 down vote accepted

$(a,b]$ is an interval from $a$ (exclusive - i.e. not including $a$ itself) to $b$ (inclusive)

$[a,b)$ is an interval from $a$ (inclusive) to $b$ (exclusive - i.e. not including $b$ itself)

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For $(a,b]$ I say the interval from $a$ to $b$ including $b$ but not including $a$.

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+1 This is the easiest to understand for non-math people. – Mark Lakata Sep 6 '13 at 17:02

For $(a,b]$ I usually say "left-open interval a b". I'm not sure whether it is grammatically and semantically correct, but it is short and everyone understands.

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"Everyone understands" -- one can't assume that in undergraduate, lower-division math classes, esp. with non-majors in attendance. – Roy Tinker Sep 6 '13 at 16:41
@Roy Tinker OK, agreed. Let's change it to "everyone who is capable of understanding anything at all understands it after hearing the explanation of what it means just 3 times" :-) – fedja Sep 6 '13 at 16:55

The interval from $a$ to $b$, left-inclusive?

In general, I have no problem with a few more words for additional clarity. So 'the interval $a$, $b$, including the point $a$' seems fine to me too.

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is there a standard to follow? – yiyi Sep 6 '13 at 7:45
@yiyi Wikipedia has a page about international standard ISO 31-31: mathematical signs and symbols for use in physical sciences and technology. In the Sets section it says that the verbal equivalent of (a,b] is "left half-open interval in ℝ from a (excluded) to b (included)" and the verbal equivalent of [a,b) is "right half-open interval in ℝ from a (included) to b (excluded)". – Iain Elder Sep 6 '13 at 13:27

I would call $(a,b]$ the left open interval and $[a,b)$ the right open interval.

(Seems like I have the subconcious implicit assumtion that "normal intervals are closed"…)

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Same proposal by @fedja a few seconds earlier... – Dirk Sep 6 '13 at 10:36
implicit assumtion that "normal intervals are closed" I rather make an assumption that if I say something like "a room with chairs in the northeast and southwest corners", then it sort of suggests that there isn't a chair anywhere else, so I can just as easily say "right-closed" instead of "left-open" but the basic idea is the same. – fedja Sep 6 '13 at 11:00
Good point. But I built my impression on the fact that I do not use "right-closed" and "left-closed" at all. – Dirk Sep 8 '13 at 9:09

$(a, b]$ is:

The interval $a$ to $b$ open on $a$ and closed on $b$

The interval $x$ such that $x$ is greater than $a$ and less than or equal to $b$.

Or any of the other answers given would be acceptable.

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Although this is just a slight variation of the already posted pronunciations, I tend to say 'half-open interval from $a$ to $b$ without $a$' for $(a,b]$ and '(..)without $b$' for $[a,b)$

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