So, in the definition of what is a square root,
$\sqrt{x}$ are all numbers $y$ such that $y×y=x$.
are there any logical mathematical symbols so that the above definition can be written using logical operators only, and no natural language?
Where can I get some introductory or reference material on all such logical symbols?
update: I noticed, some time after asking the question that the definition of square root I am giving is wrong. The square root of $x$ is to defined to be the non-negative number $y$ that satisfies $y*y=x$. But the question was about notation, not square roots, so I am leaving it as it stands due to some answers using the supplied (erroneous) definition.
4 Answers
You could write this in a few different ways... I'm not sure what you're asking, so let me show you a couple.
For one, you could define the condition $y\in\text{Sqrt}(x)$, rather than the set itself: $$ y\in\text{Sqrt}(x)\Leftrightarrow y^2=x $$
The following two are commonly used in set definitions: $$ \text{Sqrt}(x)=\{y\mid y^2=x\}\qquad \text{or}\qquad \text{Sqrt}(x)=\{y:\ y^2=x\} $$
I also see people use (and have used myself) "s.t." as an abbreviation for such that in formulas.
Usually, there doesn't need to be a symbol other than a colon or $\mid$ for "such that."
The English language version of your statement seems to describe $\sqrt x$ as a set. You could write this as:
$$y\in \sqrt{x} \iff y\in\mathbb R \land y\cdot y = x$$
Note, I've added the $y\in\mathbb R$ because you need to know the domain in which you are working. You could chaange that, of course.
This is often abbreviated as:
$$\sqrt{x} =\{y\in\mathbb R\mid y\cdot y = x\}$$
Roughly, the $\mid$ character functions as a "such that" symbol here. Sometimes a $:$ symbol is used instead.
I think I remember that I have seen notations such as $$ \sqrt x :=\iota y (y\ge 0\land y^2=x)$$ i.e. $\iota v \Phi$ is used to denote the unique element of the (hopefully) singleton set $\{v\mid \Phi\}$. While having such a notation may be useful for extreme formality, I am personally no friend of it.
I ALSO have seen a backwards ∈ symbol for "such that." I saw it in logical notation for the definition of the limit of a function. M. Del Nero
