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 $\Epsilon {\restriction_S} = \left({S, S, \in_S}\right)$, where:

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

Also see