# Membership Relation is Not Reflexive

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## Theorem

Let $\Bbb S$ be a set of sets in the context of pure set theory

Let $\mathcal R$ denote the membership relation on $\Bbb S$:

- $\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \mathcal R \iff a \in b$

$\mathcal R$ is not in general a reflexive relation.

## Proof

In the extreme pathological edge case:

- $S = \set S$

it is seen that:

- $S \in \S$

and so:

- $\forall x \in S: x \in x$

demonstrating that $\mathcal R$ is reflexive in this specific case.

However, in this case $\set S$ is a set on which the Axiom of Foundation does not apply.

This is seen in Set is Not Element of Itself.

Hence this set is not supported by Zermelo-Fraenkel set theory.

Consider the set:

- $T = \set {\O, \set \O}$

Then we immediately see that:

- $\O \notin \O$

and so $\mathcal R$ is seen to be not reflexive.

$\blacksquare$

## Sources

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations: Exercise $1.5.1$