We hear a lot about equality, diversity and inclusion now. Perhaps it would be good to remind ourselves of the formal definition.
- A and B are equal if, for all x, we have (x∈A) ↔ (x∈B).
- A and B are diverse if this is not the case; that is, there is an x with either (x∈A) but not (x∈B), or (x∈B) but not (x∈A). This used to be called “inequality”, but the term is now deprecated.
- A includes B if, for all x, we have (x∈B) → (x∈A). The older terms “subset” and “superset” have overtones of class and should be avoided.
Please note that all the above are binary. This is an obvious shortcoming: there is a high-level commission of logicians working on a non-binary version, but it is proving to be a challenging problem.
I suppose the difficulty is that although it is easy to define `binary’, it is not so easy to define `non-binary’ if there is no universal agreement about a universal set.