Definition talk:Real Interval

I've been wondering what others think about the double dots notation. Is this common enough in texts these days that the average person who is interested enough in math to be using proofwiki would recognize it? --Cynic-(talk) 21:48, 26 December 2008 (UTC)

I'd say stick with using a comma. --Joe 21:56, 26 December 2008 (UTC)

I'd say whatever you use, make it obvious what it means. It's easy enough to get confused when reading $(x, y)$ to think it may mean a coordinate pair, but (once explained) the two-dots notation is unambiguous.

When you're writing a proof, it's usually obvious to a writer what all the symbols mean, but I don't think it's reasonable to expect anyone coming upon a page from cold should have any "prior experience" of anything in the proof at all.

As for two-dots v. comma, if we are going to start using all the most up-to-date notation, then we're going to use two dots. If we're not going to be consistently using the most up-to-date notation, we should be able to continue to use, for example, $I_S$ for the identity mapping on the set $S$ rather than the horrible $1_S$, and carry on using $e$ for the identity of a monoid rather than the equally horrible $0$ or $1$, which are currently fashionable, apparently. --Matt Westwood 23:24, 26 December 2008 (UTC)

I noticed something when I was going through this: Suppose $a, b \in \R$ comes before the ToC, which isn't good. Any thoughts on how to fix this? --Cynic-(talk) 02:34, 27 December 2008 (UTC)

How's that? --Matt Westwood 11:33, 27 December 2008 (UTC)

Much better, thanks! --Cynic-(talk) 17:53, 27 December 2008 (UTC)

On the definition of real intervals
Isn't it much more elegant to define intervals as in Interval Defined by Betweenness, rather than define it as one of 10 (!) types? This avoids as well to use the vague word "between" in the sentence
 * The set of all real numbers between any two given real numbers $a$ and $b$ is called a (real) interval.

From the informal definition, it is not clear whether "between" includes endpoints. Aren't informal definitions meant to make things more understandable? :)

I could write out a proposal here on the talk page for an alternative organization. What do you think? --barto (talk) 10:19, 28 January 2017 (EST)


 * Well of course you don't know from the informal definition what sort of interval you are talking about, because "betweenness" applies to all intervals. Which is all very well, but most cases you do need to know what kind of interval you are talking about. Hence the 10 different types of interval. --prime mover (talk) 12:00, 28 January 2017 (EST)


 * I agree we should define each of the types, but what I said is I wouldn't take them as definition of "interval". I.e. rather than "A real interval is any subset of $\R$ of a type listed below." I propose to define it like: "An interval is a subset $S$ of the real numbers with the property that $x,y\in S$, $z\in\R$ and $x<z<y\implies z\in S$." And then say: "In Interval Defined by Betweenness, it is shown that every interval is of one of the types described below." --barto (talk) 12:10, 28 January 2017 (EST)


 * This way, we have a concise but formal definition, rather than a definition that's 200 lines long :) --barto (talk) 12:13, 28 January 2017 (EST)