# Confusion about the notation of empty interval

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.

• Because $(b,a) = \{ r \in \mathbb R \mid x < a < b <x \}$. – Mauro ALLEGRANZA Jan 16 '19 at 9:45
• See also the post Why is the empty set considered an interval? – Mauro ALLEGRANZA Jan 16 '19 at 9:50
• Yes, I've seen it but why's there nothing after the $≤$ in (A,≤) in the first line ? – user8718165 Jan 16 '19 at 10:06
• $(A, \le)$ 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 $\mathbb R$. – Mauro ALLEGRANZA Jan 16 '19 at 10:18
• 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. – user8718165 Jan 16 '19 at 10:27

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.