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What does the $\ni$ (backwards element of) symbol mean? It doesn't appear in the Wikipedia list of mathematical symbols, and a Google search for "backwards element of" or "backwards epsilon" turns up contradictory (or unreliable) information.

It seems it can mean both "such that", or "contains as an element". Is this correct, and if so, which is the more common usage?

Channel72
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6 Answers6

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The backwards epsilon notation for "such that" was introduced by Peano in 1898, e.g. from Jeff Miller's Earliest Uses of Various Mathematical Symbols:

Such that. According to Julio González Cabillón, Peano introduced the backwards lower-case epsilon for "such that" in "Formulaire de Mathematiques vol. II, #2" (p. iv, 1898).

Peano introduced the backwards lower-case epsilon for "such that" in his 1889 "Principles of arithmetic, presented by a new method," according van Heijenoort's From Frege to Gödel: A Source Book in Mathematical Logic, 1879--1931 [Judy Green].

Community
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Bill Dubuque
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I believe the most common usage is "such that".

This seems to agree.

While this pdf seems to indicate a growing trend in using it as "contains as an element".

Historically, it was first used to mean "such that" (see the second pdf link above).

gen-ℤ ready to perish
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Aryabhata
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    Its usage as "contains as an element" is as redundant as having both $<$ and $>$... – Mariano Suárez-Álvarez Dec 24 '10 at 18:27
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    in my experience its used more often in the "element of" context and I don't find it redundant. Here are two useful examples: (1) let $U \ni x$ be open. (2) It's also nice to use when defining or referring to a function as in, $A \ni a \mapsto f(a) \in B$. – Eric O. Korman Dec 24 '10 at 18:30
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I have always used $\ni$ to represent "such that." Many times people do not use this notation though; rather, they use s.t. or just say "such that."

Also, you can see that Wikipedia uses the ":" notation, which is also correct; however, I usually only use that notation when in curly brackets (ie { }).

You can visit [1] to see an actual discussion of it in a list of mathematical symbols.

This was a very good question, by the way.

[1] http://www.math.ucdavis.edu/~anne/WQ2007/mat67-Common_Math_Symbols.pdf

Tyler Clark
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It is for example used for functions that are inconvenient to refer to with a symbol. E.g. if $G$ is a group, then the composition function can be referred to like this:

$$ G \times G \owns (x,y) \mapsto xy \in G $$

The fact that the LaTeX command for it is \owns should also give a clue as to what it's used for...

kahen
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The fact that \ni is \in written in the opposite direction, should also support the idea of those who think that this symbol means "contains as an element".

Miles Eagle
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I believe there are two similar symbols, $\ni$ (named \ni) and $\backepsilon$ (named \backepsilon). The former is similar to $\in$ (named \in), meaning 'contains' or 'belongs to', but is more conveniently used in text lines, while the latter is the true 'such that.'

And I found an example on page 196 of A gentle introduction to group theory, where the orbits of permutation are defined，

$$ i\cong_{\phi} j \Leftrightarrow \exists\, m\in\mathbb{Z}\backepsilon j=\phi^m(i). $$

This is exactly why I am here to answer this question, especially since I haven't found a satisfactory answer.

Dong Chen
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