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 $\Epsilon {\restriction_S} = \struct {S, S, \in_S}$, 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.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.22$