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Given an open interval (or closed it makes little difference to the question) on the Real Field $(a, b)$, where $a,b$ are real numbers, and an arbitrary predicate $P(x)$, which is true for all elements of the interval.

Using Set Theory Notation, I can represent what I've described in the following way.

$$P(x) \ \forall \ x \in \{x \ | \ x \in \mathbb{R} \ \land \ a < x < b \}$$

Using Interval Notation, however what is the correct way to translate from Set Theory to Interval Notation. Which of the following options would you say are correct? (no this is NOT homework, these are a list of different commonly used styles of interval notation, I've come across)

  1. $P(x) \ \forall x \in \mathbb{R} \ (a, b)$

  2. $P(x) \ \forall x \in \mathbb{R} \in (a, b)$

  3. $P(x) \ \forall x \in (a, b)$


Option 1, is what I tend to commonly use, but I've come to realize that it may cause confusion as "$\forall x \in \mathbb{R}$" is commonly used to represent the interval $(-\infty, \infty)$.

Option 2 seems to be the most explicit, as it shows that $x$ is an element of the Real Field, and lies within the interval $(a, b)$. Furthermore it can be used to construct more complicated statements such as the following :

$P(x) \ \forall x \in \mathbb{Q} \in (a, b)$, which would mean a predicate would hold true for all rational numbers within the interval on the Real Field, between the real numbers $a, b$

Option 3 is what I see used most often, however it seems to be the least explicit, as it does not even denote the Set (Field in this case), which $x$ belongs do. Yes, it says that $x$ is an element of the interval $(a, b)$, but $x$ could be a complex number, integer, rational, irrational or real, as we haven't specified explicitly, which field $x$ is an element of.

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Option 3 is by far the most common, as you say. It is almost always used to refer to $\{ x \in \mathbb{R} : a < x < b \}$. It doesn't even make sense to consider an interval in $\mathbb{C}$, because there is no order on $\mathbb{C}$ that respects the field structure. Your Option 1 is so non-standard that I would ask for clarification if I saw it and had the option of asking for clarification. Option 2 is written in such a way that it appears false: $\mathbb{R} \not \in (a,b)$.

In the absence of further clarification, $x \in (a,b)$ essentially always means $a < x < b$ and $x \in \mathbb{R}$. If you wish to say something else, you might say "$x \in \mathbb{Q} \cap (a,b)$" or whatever, and of course if you're working in a different total order than $(\mathbb{R}, <)$ then the interval notation will refer to intervals in that order.

There are special cases where you can emphasise that you're in $\mathbb{R}$: it's fairly close to standard to write $\mathbb{R}^{>0}$ for the strictly positive reals, for instance. Usually best to stick to $\{x \in \mathbb{R}: a<x<b\}$ if you need to be explicit, though.

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