# 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: $$\mathbf{\omega} = \iiint_A f(x,y,z)\, \mathrm{d}x \mathrm{d}y \mathrm{d}z$$$$\mathbf{\upsilon} = \frac{1}{2}\oint\nabla \cdot f(x,y,0)$$$$\epsilon_{1}^{2} = \left(\lim\limits_{x\rightarrow-\infty} f(x,y,0)\right)^2 + \left(\lim\limits_{y\rightarrow-\infty} f(x,y,0)\right)^2$$$$\epsilon_{2}^{2} =\left(\lim\limits_{x\rightarrow\infty} f(x,y,0)\right)^2 + \left(\lim\limits_{y\rightarrow\infty} f(x,y,0)\right)^2$$$$\omega,\upsilon \in [\epsilon_1,\epsilon_2]$$

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Try adding an edit to the OP to clarify things further :) –  Shaun Nov 11 '13 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. –  roman Nov 11 '13 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. –  Shaun Nov 11 '13 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. –  NictraSavios Nov 11 '13 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. –  NictraSavios Nov 11 '13 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 $$\scriptstyle \forall e:\bigl[(e\in A)\Leftrightarrow \exists x:\exists y:(((((((x\in\mathbb R)\land(y\in\mathbb R))\land (e=\langle x,y\rangle)) \land(x\ge 0))\land(1\ge x))\land(y\ge 0))\land(1\ge y))\bigr]$$

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 $x,y\in[0,1]$.

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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? –  NictraSavios Nov 11 '13 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. –  Henning Makholm Nov 11 '13 at 13:17
So then what is the simplest set-builder notation for what I'd like to describe? –  NictraSavios Nov 11 '13 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? –  Henning Makholm Nov 11 '13 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. –  NictraSavios Nov 11 '13 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.

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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. –  NictraSavios Nov 11 '13 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. –  Shaun Nov 11 '13 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". –  NictraSavios Nov 11 '13 at 12:34
But it's enough. You're fine with just that. Relax. –  Shaun Nov 11 '13 at 12:37
We could be talking past each other. Would anyone care to step in and clarify things? :) –  Shaun Nov 11 '13 at 12:38

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

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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 $\leq$? –  NictraSavios Nov 11 '13 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? –  Taladris Nov 12 '13 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. –  NictraSavios Nov 12 '13 at 7:43

We have the following

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

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

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I'm looking for more formal, rigorous notation involving both x and y –  NictraSavios Nov 11 '13 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. –  Malice Vidrine Nov 11 '13 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. –  NictraSavios Nov 11 '13 at 13:30