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.

See also the post Why is the empty set considered an interval? — Mauro ALLEGRANZA, Jan 16 '19 at 09:50
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
Obviously, a non-empty interval means that the said interval cannot be a "degenerate" one, like $(a,a)$ and $(b,a)$ with $b > a$, because in these cases the inetrvals are empty. — Mauro ALLEGRANZA, Jan 16 '19 at 10:31
See Interval (mathematics) : open $(a,b)$; closed $[a,b]$. — Mauro ALLEGRANZA, Jan 16 '19 at 10:40

score 3 · Accepted Answer · answered Jan 16 '19 at 09:47

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.

Confusion about the notation of empty interval

1 Answers1