Set Equivalence behaves like Equivalence Relation/Warning

Set Equivalence behaves like Equivalence Relation: Warning

It has been shown that set equivalence exhibits the same properties as an equivalence relation.

However, it is important to note that set equivalence is not strictly speaking a relation.

This is because the collection of all sets is itself specifically not a set, but a class.

Hence it is incorrect to refer to $\sim$ as an equivalence relation, although it is useful to be able to consider it as behaving like an equivalence relation.

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $3$: Cardinality: Exercise $1 \ \text{(c)}$