# 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**.

- $\Epsilon := \left\{{ \left({ x , y }\right) : x \in y }\right\}$

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

It consists of precisely those ordered pairs $\left({ x , y }\right)$ satisfying $x \in y$.

The behavior 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

- Epsilon is Foundational
- The definition of standard structure

## Sources

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