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Currently, I'm studying about intervals. I got the basic understanding and knowledge about them but while reading the page about intervals in Wikipedia, under the classification of intervals tab, I didn't understand some of the notations used for the empty interval. They are

$(b,a)$

$(b,a]$

$[b,a)$

where $b>a$

What does the upper-bound (b) written before the lower-bound (a) in these three notations mean? And how do these notations work?

Thank you for your valuable suggestions.

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    $\begingroup$ Because $(b,a) = \{ r \in \mathbb R \mid x < a < b <x \}$. $\endgroup$ – Mauro ALLEGRANZA Jan 16 '19 at 9:45
  • $\begingroup$ See also the post Why is the empty set considered an interval? $\endgroup$ – Mauro ALLEGRANZA Jan 16 '19 at 9:50
  • $\begingroup$ Yes, I've seen it but why's there nothing after the $≤$ in (A,≤) in the first line ? $\endgroup$ – user8718165 Jan 16 '19 at 10:06
  • $\begingroup$ $(A, \le)$ is not an "interval"... It is the "environment"; in calculus it is $\mathbb R$ with the usual ordering $\le$ between real numbers. An interval $(a,b)$ or $[a,b]$ is a subset of $\mathbb R$. $\endgroup$ – Mauro ALLEGRANZA Jan 16 '19 at 10:18
  • $\begingroup$ Thank you so much!! the domain is a set of real numbers that contains a non-empty open interval could you please tell me what the highlighted line means? This is the last thing I'd like to know. $\endgroup$ – user8718165 Jan 16 '19 at 10:27
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The empty interval is obviously ... empty; it is an empty set of numbers.

Why e.g. $(b,a) = \emptyset$ when $b > a$ ?

Because, in general :

$(a,b) = \{ x \in \mathbb R \mid a < x \text { and } x < b \}$.

When $b > a$ we have that :

$(b,a) = \{ x \in \mathbb R \mid b < x \text { and } x < a \}$

and there are no $x$ that satisfies the condition.

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