# Correct formal interval notation

I can't find any definitive answer on this topic, maybe that's because there isn't one, but I figured if there was a place to ask then SE was it!

To describe a set in which $x$ and $y$ are in the interval $[0,1]$ formally, would one write $\{x,y \in \mathbb{R} | 0 \leq x,y \leq 1\}$ or should the terms be split up into separate notations, or have a specific symbol between them? Also, for a short form of this relation, does $x,y \in [0,1]$ suffice, or is there a better notation?

# Edit

Basically, I'm asking for an notation that makes $\{x \in \mathbb{R} | 0 \leq x \leq 1 \}$ and $\{y \in \mathbb{R} | 0 \leq y \leq 1 \}$ into one expression.

# Second Edit

In context, what I'm trying to represent is: $$\mathbf{\omega} = \iiint_A f(x,y,z)\, \mathrm{d}x \mathrm{d}y \mathrm{d}z$$$$\mathbf{\upsilon} = \frac{1}{2}\oint\nabla \cdot f(x,y,0)$$$$\epsilon_{1}^{2} = \left(\lim\limits_{x\rightarrow-\infty} f(x,y,0)\right)^2 + \left(\lim\limits_{y\rightarrow-\infty} f(x,y,0)\right)^2$$$$\epsilon_{2}^{2} =\left(\lim\limits_{x\rightarrow\infty} f(x,y,0)\right)^2 + \left(\lim\limits_{y\rightarrow\infty} f(x,y,0)\right)^2$$$$\omega,\upsilon \in [\epsilon_1,\epsilon_2]$$

• Try adding an edit to the OP to clarify things further :) Nov 11, 2013 at 12:53
• I still don't get what he's asking for, given that he just dismissed to perfect answers for two valid interpretations of his question. Nov 11, 2013 at 12:55
• I suppose to be completely formal (and pedantic) you could take the route of Russell & Whitehead in their Principia Mathematica, but that might be extreme overkill. It depends on your needs. Keeping to a chosen convention suffices for most purposes. See Henning's answer. Nov 11, 2013 at 12:58
• I consider one of them valid... the one I commented on asking for a more rigorous one. Bassically, I'm asking for an notation that makes $\{x \in \mathbb{R} | 0 \leq x \leq 1 \}$ and $\{y \in \mathbb{R} | 0 \leq y \leq 1 \}$ into one expression. Nov 11, 2013 at 13:04
• @BrianM.Scott You are correct, I was reading someone elses answer while updating the document and typed what I was reading without considering its logical meaning. Sorry about that. Updated. Nov 11, 2013 at 13:32

If you want to be completely formal, go to first-order logic and axiomatic set theory and consider the set $A$ defined by the property $$\forall e:\bigl[(e\in A)\Leftrightarrow \exists x:\exists y:(((((((x\in\mathbb R)\land(y\in\mathbb R))\land (e=\langle x,y\rangle)) \land(x\ge 0))\land(1\ge x))\land(y\ge 0))\land(1\ge y))\bigr]$$

But wait! $\mathbb R$ and $e=\langle x,y\rangle$ and $\ge$ are all abbreviations of formulas of several lines each, so you need to expand them. And depending on the flavor of logic you're working in, $\Leftrightarrow$ as well as $\exists$ or $\forall$ may also be abbreviations that you have to expand ...

... you don't want to be completely formal. Writing mathematics down is a matter of communicating your ideas to human readers, not formal systems. Whichever notation you use that will convey your ideas unambiguously and succinctly is right -- and "most formal notation possible" is very rarely a worthwhile goal.

You can write $\{x,y\}\subseteq [0,1]$ if you want, but in practice that will just confuse readers (because this notation makes it looks like the particular set $\{x,y\}$ is conceptually relevant to what you're trying to say, which it isn't) and will bring you no benefit compared to the more informal $x,y\in[0,1]$.

• Would something like $S = \{x,y\}$ followed by $\{S \in \mathbb{R} : 0 \leq S \leq 1\}$ be appropriate? Or does this mean something else, like the cartesian cross product mentioned below? Nov 11, 2013 at 13:16
• @NictraSavios: That would just amount to even more amounts of confusing fluff that doesn't help any reader grasp what you're saying. (It also doesn't make sense formally, but that isn't even the point). Saying something in a complex way when it can be said simply is not a mathematical virtue. Nov 11, 2013 at 13:17
• So then what is the simplest set-builder notation for what I'd like to describe? Nov 11, 2013 at 13:21
• @NictraSavios: You still haven't explained why you're not satisfied with $x,y\in[0,1]$. Is it too easy to understand for your taste? Nov 11, 2013 at 13:22
• Well, In context (I edited the OP reluctantly), I want to be able to forgo the $\epsilon$ terms and simply write the limits in the set-builder notation. My reason for doing so is because I find it hard to understand myself. Nov 11, 2013 at 13:27

A partial answer: there's no such thing as "correct" notation. Go with whatever helps you to understand what you're doing, and make sure you be explicit about it.

But you're best off sticking to the more conventional suggestions you'll no doubt get from others here.

• I do understand that, infact I usually use whatever notation helps me the best, but in this case I want the most formal, rigorous notation. Nov 11, 2013 at 12:28
• No: that's the thing. There is no "most formal" way; not strictly speaking anyway: there's just a choice of conventional notations. Nov 11, 2013 at 12:31
• Again, I know... I suck with words. A more rigorous and clear notation than $x,y \in [0,1]$ then, preferably one which conforms to the conventional formal interval notation "variable in general number set such that variable is between two values of said number set". Nov 11, 2013 at 12:34
• But it's enough. You're fine with just that. Relax. Nov 11, 2013 at 12:37
• We could be talking past each other. Would anyone care to step in and clarify things? :) Nov 11, 2013 at 12:38

$E=\left\{(x,y)\ \mid\ 0\leqslant x,y\leqslant 1\ \right\}$ is not an interval but the Cartesian product $[0,1]\times[0,1]$.

• Shouldn't this be a comment, and not an answer? Also... not very helpful in offering a solution. Thank you though. One quirky question though, why $\leqslant$ over $\leq$? Nov 11, 2013 at 12:26
• Sorry but it is actually an answer: $x,y\in[0,1]$ and $(x,y)\in[0,1]\times[0,1]$ have exactly the same meaning. Being helpful or not is a matter of taste. Using $\leqslant$ instead of $\leq$ is also a matter of taste. When I write inequalities, I use $\leqslant$, so why not doing the same with Latex? Nov 12, 2013 at 5:54
• Actually, the original question asked "should the terms be split up into separate notations, or have a specific symbol between them?" ... and this answers none of that. Nov 12, 2013 at 7:43

We have the following

$[0,1]=\{x\in\mathbb{R} | 0\leq x\leq1\}$.

Then you can simply state that numbers $y,z\in[0,1]$.

• I'm looking for more formal, rigorous notation involving both x and y Nov 11, 2013 at 12:27
• @NictraSavios: Um, this is pretty standard formal notation. What is the context this comes up in? The things that you're asking to combine are lexical variants of notation for the same set. Combining them makes no sense. Nov 11, 2013 at 13:13
• Well, to me it feels wrong to state that the interval of y relies on x being in that interval. I'd like them to seem... "independent" of each other, yet equal. Nov 11, 2013 at 13:30