Correct formal interval notation

Question

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

Answer 1

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.

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

Answer 2

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.

Answer 3

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

Answer 4

We have the following

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

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