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So basically, interval notation has 4 basic forms

1) $\{1,2,3,4\}$ with example: the set of natural number $N=\{1,2,3,4,5,6,7,8,9\}$

2) $(a,b)$

3) $[a,b)$ or $(a,b]$

4) $[a,b]$

The question is: "how do I express the set of real numbers $\mathbb{R}$ in terms of interval notation?"

I've read a lot of books but no books about I read actually express real numbers in terms of interval notation. I assume that the right way to express the set of real number $\mathbb{R}$ is: "$\mathbb{R}=(-\infty,\infty)$" (a letter, then the equal symbol "=", and the expression after that), but really I have no documents to confirm the accuracy of this expression for real number.

Could anyone tell me if I am expressing correctly?

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  • $\begingroup$ I would disagree that $\{1,2,3,4\}$ is in "interval notation." I would call that something else entirely. As to the question of the set of the real numbers, yes $\Bbb R=(-\infty,\infty)$ where $(-\infty,\infty)$ is taken to be interpreted as an interval. The definition for an interval $(a,b)$ is the set of real numbers that are strictly larger than $a$ and strictly less than $b$. That is to say, $(a,b)=\{x\in\Bbb R~:~a<x<b\}$. Since all real numbers satisfy $-\infty<x<\infty$, we get our desired result. That being said, I still prefer to call it $\Bbb R$. $\endgroup$ – JMoravitz Nov 11 '17 at 4:02
  • $\begingroup$ Thank you so much JMoravitz for explaining to me of how {a,b,c,..} is different from (a,b), [a,b), or [a,b]. Could I ask you where (which book) I can find the confirmation of R=(−∞,∞) or R=−∞<x<∞ ? I totally understand your answer. The problem I have is within my 5 textbooks, they never write anything similar to R=(−∞,∞) $\endgroup$ – PandoraU.U.D Nov 11 '17 at 5:07
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    $\begingroup$ Well, it is circular. In order to define an interval, we need to first have defined the real numbers, so $\Bbb R=(-\infty,\infty)$ is not and cannot be a definition. Assuming the book defines the reals elsewhere, there is no reason to introduce an additional notation after it. $\endgroup$ – JMoravitz Nov 11 '17 at 5:16
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A common misconception beginning mathematicians have is that mathematical notation follows some universal and inviolate rules; perhaps this stems from the fact that mathematical arguments follow rigorous rules of inference.

Notation is all about human communication. It is "right" if your readers can easily understand what you are trying to say and "wrong" if it is incomplete, confusing, or misleading. Clashing with some common conventions is one way of being confusing, so it's good to familiarize yourself with the most common ways others write things, but keep in mind that different mathematicians and authors will use different notation and that there is no "one right way" to write mathematics.

With that in mind, I would say that you have identified two common ways to define sets: the first as an explicit enumeration of elements, $$S = \{1, 2, 5\}.$$ You can write (small) intervals of natural numbers this way, but it is not so convenient to define intervals of real numbers like this, since you cannot list the uncountably infinite real numbers between the endpoints $a$ and $b$ of an interval. The above, though, is a special case of set-builder notation $$S = \{x\in\mathbb{R}\ \mathrm{s.t.}\ \textrm{some condition on } x \};$$ using this notation you can write an open interval as $$S = \{x\in\mathbb{R}\ \mathrm{s.t.}\ a < x < b\}.$$ Set-builder notation is extremely powerful and common, but notice that again there is no universal "right" notation; some authors will write ":" or "|" instead of "s.t", etc.

The second set of examples you list are common shortcuts for writing intervals, using $()$ for open endpoints and $[]$ for closed ones. Notice that some authors write $]a,b[$ for the open interval instead of $(a,b)$, but yours is probably the most common convention. In this notation $(-\infty, \infty)$ would indeed indicate the set of all real numbers, although you should be aware that this notation is not complete free of potential confusion: is this an interval of real numbers, rational numbers, integers, or something else? In context it might be obvious, but there is a potential ambiguity.

Like your books, I wouldn't try to write the set of real numbers using an interval at all: this is a bit circular, since intervals are subsets of the real numbers, and in any case it's quite safe to assume that your readers know what the real numbers are without being handed an explicit set. Actually defining the real numbers rigorously is a surprisingly subtle problem, and if you are interested one keyword for starting your research is "Dedekind cuts."

For saying that a variable comes from the real numbers, I would write "$x\in\mathbb{R}$" or "a real number $x$" instead of "$x\in (-\infty,\infty)$" in most situations.

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  • $\begingroup$ Alright thank you so much user7530 for making everything clear. According to your suggestion, when I want to define R by using symbols, I should use: "R={x∈R | −∞<x<∞} rather than R=(-∞,∞) ? $\endgroup$ – PandoraU.U.D Nov 11 '17 at 5:10
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    $\begingroup$ Why do you want to define $\mathbb{R}$ using symbols? $\mathbb{R}$ is already a standard symbol. $\endgroup$ – user7530 Nov 11 '17 at 6:15
  • $\begingroup$ Originally, I can see, in all known math textbooks, that they define other set of numbers such as the set of natural number (N={1,2,3,4,5,6,7,8,9}), set of integers (J={...-2,-1,0,1,2,...}). The only set they don't define by using symbols are Real numbers and Complex numbers. I only know that we can say x∈(−∞,∞) and x∈R. But I never see something written as R=(−∞,∞) $\endgroup$ – PandoraU.U.D Nov 14 '17 at 4:11
  • $\begingroup$ @PandoraU.U.D Those aren't definitions, they're more like reminders about names and convention (like whether $\mathbb{N}$ includes zero). You can't define the integers just by giving a list of elements; you have to also specify its algebraic properties (how integers add and multiply). $\endgroup$ – user7530 Nov 14 '17 at 4:39

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