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.
Recently I learned that equality can be achieved via repeated cancellation and that this method is both necessary and sufficient. It therefore seems possible to define equality in a unary fashion until a critical part remains on the left and the right hand side is empty.
You need (binary) inclusion to define the empty set, right?
Please don’t destroy your lectures notes so as to avoid “subset”.