# Backwards epsilon

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?

• Just element of, in the other direction. – Jonas Teuwen Dec 24 '10 at 18:21
• By the way, I am of the opinion that $\in$ is not an epsilon! – Mariano Suárez-Álvarez Dec 24 '10 at 18:29
• Mariano, the $\in$ is usually called "epsilon relation" in the framework of set theory, and it stemmed from Tarski's $\varepsilon$ notation, as far as I know. – Asaf Karagila Dec 24 '10 at 19:36
• Three years later; now I know that the $\varepsilon$ symbol was in fact due to Peano, and not Tarski. – Asaf Karagila Dec 29 '13 at 14:07

I believe the most common usage is "such that".

This seems to agree: http://www.math.ucdavis.edu/~anne/WQ2007/mat67-Common_Math_Symbols.pdf

While this pdf here: http://faculty.ssfs.org/~doschel/2008-2009%20School%20Year/Calculus/Math%20Symbols.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).

• Its usage as "contains as an element" is as redundant as having both $<$ and $>$... – Mariano Suárez-Álvarez Dec 24 '10 at 18:27
• 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

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].

• Great minds are just as capable as the other ones of introducing bad notation :) – Mariano Suárez-Álvarez Dec 24 '10 at 18:46

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...

• \ni also works (contrast with \in). – Eric O. Korman Dec 24 '10 at 18:42

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.

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".