Definition:Epsilon Relation

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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.

In class-builder notation:

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

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 $\Epsilon {\restriction_S} = \struct {S, S, \in_S}$, where:

${\in_S} := \set {\tuple {x, y} \in S \times S: x \in y}$

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.