# “such that” logical symbol

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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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. – akinn Jun 29 '13 at 13:43
I used to sometimes see a backwards $\in$ symbol or something like it for "such that." – Thomas Andrews Jun 29 '13 at 13:46
@Andrews I thought the reverse Greek epsilon was for 'exists' – Marcus Junius Brutus Jun 29 '13 at 13:47
"Exists" is usually a reversed (Roman) E: $\exists$ – Nick Peterson Jun 29 '13 at 13:47
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? – joeA Jun 29 '13 at 15:10

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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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 $\frac{1}{y}=x$. – user26486 Sep 2 '14 at 20:17
in all my years in methematics i never saw that "ambiguity". in latex, \mid is even a different symbol than | for "divides". – peter Apr 24 '15 at 21:57

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 frined of it.

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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. – Marcus Junius Brutus Jun 29 '13 at 14:18
That $\iota$ notation is from Whitehead and Russell's Principia Mathematica. – MJD Jul 2 '13 at 17:14

Usually, there doesn't need to be a symbol other than a colon or $\mid$ for "such that."

Then 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 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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