Confusion about the notation of empty interval

Question

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.

Answer 1

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.