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It's been I couple of years since I learned intervals and I've forgotten enough of it for it to confuse me on a daily basis.

I have 2 questions:

  1. What is the difference between [0, 1) and [0, 1]. Is the first continuous (from 0 to 0.99...9) and the other discrete?

  2. Can discrete values be represented on a real number line? In 9 minus 6 (image here) for example, it seems like I'm counting the spaces between the vertical lines. If I'm doing right, then the discrete values become continuous. Am I understanding this incorrectly?

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    $[0, 1)=\{0 \leq x < 1\}$ while $[0,1]=\{0\leq x \leq 1\}$. – 2017-01-22
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    I understand this, but does this imply that instead of 1, it is 0.99 continuous? – 2017-01-22
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    It turns out to be a pretty big difference. $[0,1]$ has a maximal element while $[0,1)$ does not... – 2017-01-22
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    I think there is some non-standard usage of *discrete* and *continuous* happening here, with some infinitesimal-style stuff thrown in. I wouldn't try to phrase the difference between the intervals in terms of anything like that, but I guess that's what the question is about. – 2017-01-22
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    @danm07 The difference is that $1$ belongs to $[0,1]$ while **it does not** belong to $[0, 1)$. I don't know what you mean by discrete values or 0.99 being continous. Numbers are just numbers, neither discrete nor continous. It has nothing to do with such ideas. So it's just that: one set is bigger then the other by exactly one element, namely $1$. – 2017-01-22

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$[0,1]$ is the set of numbers between $0$ and $1$, including $0$ and including $1$. (It's the set of numbers for which $“0\le x\le1”$ is true.)

$[0,1)$ is the set of numbers between $0$ and $1$, including $0$ but not including $1$. (It's the set of numbers for which $“0\le x<1”$ is true.)

Side note: I have no idea how one is meant to pronounce these.

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    Right, I understand this bit. But I guess, if I were to compare the sums of [0,1) and [0,1], would there be a difference? Or would it be discrete so as to be equal? – 2017-01-22
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    @danm07 What do you mean by the "sums"? Do you mean how many numbers are in the intervals? – 2017-01-22
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    What do you mean by sums?? This needs clarification... – 2017-01-22
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    @AkivaWeinberger Sorry. and yes, I do mean how many numbers are in the interval, if that makes any sense. (i.e. in [0,100] vs [0, 100) there are 100 and 99 respectively, if the interval was changed to [0,1] vs [0,1) how would these two compare?) – 2017-01-22
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    @danm07 The technical term for this is _cardinality_. It can be shown that the set of integers $\Bbb Z$, the set of even integers $2\Bbb Z$, and the set of rationals $\Bbb Q$ all have the same cardinality, and that the cardinality of the set of reals $\Bbb R$ is even bigger. The cardinalities of $[0,1]$ and $[0,1)$ are equal; in fact, their cardinalities are equal to the cardinality of $\Bbb R$. – 2017-01-22
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    Two sets are said to have the same cardinality if there is a _one-to-one_ and _onto_ function between them, also called a _bijection_. (Roughly speaking, this is a way of uniquely assigning every element of one to an element of another. The function $f(x)=2x$ provides a bijection between the set of integers and the set of even integers.) It is not easy to find a bijection between $[0,1]$ and $[0,1)$ — and this bijection will not be continuous — but it can be done. @danm07 – 2017-01-22
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$[0,1]$ includes $1$, while $[0,1)$ does not. However, because of how the real numbers work, you can't actually give a maximum element to the set $[0,1)$, since it's open. Remember that $0.999\ldots = 1$.

As for the second question, you have to note that the set of real numbers in any interval is an uncountable infinity, so it doesn't work the way you would hope. You can definitely choose individual points, but you can't just keep adding them on until you get everything, even in an infinite amount of time.

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    Wow... the thing about the set of real numbers in an interval is like a numerical black hole. – 2017-01-22