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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.

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    $\begingroup$ In set theory \mid or : is often used but I haven't really seen any logical symbol used for "such that" in other situations. But now that I think of it there is the \ni symbol used as "such that" in mathematical logic I believe. $\endgroup$
    – user41489
    Commented Jun 29, 2013 at 13:43
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    $\begingroup$ I used to sometimes see a backwards $\in$ symbol or something like it for "such that." $\endgroup$
    – Thomas Andrews
    Commented Jun 29, 2013 at 13:46
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    $\begingroup$ "Exists" is usually a reversed (Roman) E: $\exists$ $\endgroup$
    – Nick Peterson
    Commented Jun 29, 2013 at 13:47
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    $\begingroup$ I had a professor who used $\ni$ frequently. He wrote it quickly or in a stylized manner, and I never knew what the symbol actually was. He also wrote "suppose" as a sort of uppercase $S$ with a lowercase $p$ superimposed on the bottom half. Detexify couldn't find that one for me. Anyone seen that? $\endgroup$
    – joeA
    Commented Jun 29, 2013 at 15:10
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    $\begingroup$ Possible duplicate of Symbol for “such that” (not in set) $\endgroup$
    – Mr Pie
    Commented Jun 20, 2018 at 12:36

4 Answers 4

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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.

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  • $\begingroup$ I hate the ambiguity of the set notation. Though extremely unlikely, $\text{Sqrt}(x)=\{y\mid y^2=x\}$ can mean that the statement $y\mid y^2$ is the only element of the set $\text{Sqrt}(x)$, where in the same context we have that $x=y^2$, and $\text{Sqrt}(x)=\{y:\ y^2=x\}$ can mean that $\text{Sqrt}(x)$ is a set containing only the statement $\frac{1}{y}=x$. $\endgroup$
    – user26486
    Commented Sep 2, 2014 at 20:17
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    $\begingroup$ in all my years in methematics i never saw that "ambiguity". in latex, \mid is even a different symbol than | for "divides". $\endgroup$
    – peter
    Commented Apr 24, 2015 at 21:57
  • $\begingroup$ @peter I agree, and $:$ can get confusing when dealing with ratios. Look at my comment above. $\endgroup$
    – Mr Pie
    Commented Jun 20, 2018 at 12:36
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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.

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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.

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  • $\begingroup$ Yeah I think I made a mistake in the question in that the square root should be defined as a single number (the "principal square root"), and not as a set, so I think from a strict mathematical perspective your answer is the correct one. $\endgroup$
    – Marcus Junius Brutus
    Commented Jun 29, 2013 at 14:18
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    $\begingroup$ That $\iota$ notation is from Whitehead and Russell's Principia Mathematica. $\endgroup$
    – MJD
    Commented Jul 2, 2013 at 17:14
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    $\begingroup$ cf. planetmath.org/hilbertsvarepsilonoperator $\endgroup$
    – Manuel Eberl
    Commented Sep 24, 2016 at 13:10
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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

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  • $\begingroup$ This does not answer the question; it is merely a comment. I recommend that you revise over it and make improvements, or just delete it altogether. $\endgroup$
    – Mr Pie
    Commented Sep 8, 2018 at 7:47

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