# Definition:Epsilon Relation

## Definition

In the language of set theory $\in$, the membership primitive, is neither a class nor a set, but a primitive predicate.

To simplify formulations, it is useful to introduce a class which behaves identically to the standard membership relation $\in$ for sets.

This class, denoted $\Epsilon$, will be referred to as the **epsilon relation**.

- $\Epsilon := \set {\tuple {x, y}: x \in y}$

Thus, explicitly, $\Epsilon$ is a relation, taking arguments from ordered pairs of sets $x$ and $y$.

It consists of precisely those ordered pairs $\paren {x, y}$ satisfying $x \in y$.

The behaviour is thus seen to be identical to regular membership with sets.

It is **not** the same as class membership, because $x$ and $y$ must be set variables.

### Restriction of Epsilon Relation

Let $S$ be a set.

The **restriction of the epsilon relation** on $S$ is defined as the endorelation $\Epsilon {\restriction_S} = \left({S, S, \in_S}\right)$, where:

- $\in_S \; := \left\{{\left({x, y}\right) \in S \times S: x \in y}\right\}$

## Also see

## Historical Note

The symbol for **is an element of ** originated as $\varepsilon$, first used by Giuseppe Peano in his *Arithmetices prinicipia nova methodo exposita* of $1889$. It comes from the first letter of the Greek word meaning **is**.

The stylized version $\in$ was first used by Bertrand Russell in *Principles of Mathematics* in $1903$.

See Earliest Uses of Symbols of Set Theory and Logic in Jeff Miller's website Earliest Uses of Various Mathematical Symbols.

$x \mathop \varepsilon S$ could still be seen in works as late as 1951: Nathan Jacobson: *Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts* and 1955: John L. Kelley: *General Topology*.

Paul Halmos wrote in *Naive Set Theory* in $1960$ that:

*This version [$\epsilon$] of the Greek letter epsilon is so often used to denote belonging that its use to denote anything else is almost prohibited. Most authors relegate $\epsilon$ to its set-theoretic use forever and use $\varepsilon$ when they need the fifth letter of the Greek alphabet.*

However, since then the symbol $\in$ has been developed in such a style as to be easily distinguishable from $\epsilon$, and by the end of the $1960$s the contemporary notation was practically universal.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.22$