For a real number $x$,
$$\lfloor x\rfloor=\max\{n\in\mathbb{Z}\mid n\leq x\}.$$
I'd like to add though, that "$\lfloor x\rfloor$" is mathematical notation, just as much as the right side of the above equation is; the right side might use more "basic" constructions, but you can then ask about
$$\max,\qquad\in,\qquad \mathbb{Z},\qquad {}\mathbin{\mid}{},\qquad \leq$$
and so on. At some point you just have to start writing notation and explaining it in words and hope your readers understand. So I disagree with your phrasing of the question.
I don't understand why this seems so difficult to the other people trying to give answers. I notice the discrete-math tag.
One want to write what you want is:
$$
\forall x,y \in \mathbb{R} (xy\in \mathbb{R}\setminus\mathbb{Q} \to (x\in \mathbb{R}\setminus\mathbb{Q} \lor y \in \mathbb{R}\setminus \mathbb{Q})).
$$
In "English" this is saying that: for all $x$ and $y$ real numbers, if the product of $x$ and $y$ is a real number, but not a rational number (i.e. $xy$ is irrational), then $x$ is a real number, but not a rational number (i.e. $x$ is irrational) or $y$ is a real number but not a rational number (i.e. $y$ is irrational).
You could use $\implies$ instead of $\to$. Also, some might prefer fewer parentheses.
Some also write $\mathbb{R} - \mathbb{Q}$ instead of $\mathbb{R}\setminus\mathbb{Q}$. This is simply the set of real numbers minus the set of rational numbers. In general $A\setminus B$ is the set of elements in $A$ that are not in $B$. So $\mathbb{R}\setminus \mathbb{Q}$ is the set of irrational numbers.
As mentioned in a comment by @HenningMakholm one might also prefer to write $x\notin \mathbb{Q}$ instead of $x\in\mathbb{R}\setminus \mathbb{Q}$. This, however is only good because we gave the domain as $\mathbb{R}$.
As also mentioned in other comments, while we read $\forall$ as for all, the symbol doesn't just replace the words. The symbol has a precise (mathematical) meaning. Likewise, $\lor$ mean or, but it is used between the two statements $x\in\mathbb{R}\setminus \mathbb{Q}$ and $y\in\mathbb{R}\setminus \mathbb{Q}$.
Best Answer
\begin{align} X &= \{n : \textrm{$n$ is a positive integer greater than or equal to $10$}\} \tag{1} \\ &= \{n : n \in \mathbb Z^+ \textrm{ and } n \geq 10\} \tag{2} \\ &= \{n \in \mathbb Z^+ : n \geq 10\} \tag{3} \\ &= \{10,11,12,\dots\} \tag{4} \end{align}
$(1):$ Yes, you can use words.
$(2):$ This is the most correct.
$(3):$ The standard notation for the latter.
$(4):$ This is well understood, although it is a bit vague.