I hope I'm in the right place.
I am no mathematician - just an interested bystander - so please forgive any naivety and stupid questions.
I am going through the sections on Limits & Continuity in Russell's "Introduction To Mathematical Philosophy". One thing that has got me completely stumped at the moment is his definition of the minima of a class with respect to a relation. It goes something like this:
Given 2 classes, a & b, & a relation, P, on a&b, then the minima of a with respect to P is:
Those members of a and any members of the field of P (I'll use P.field from now on) that are not related by P from a.
Now "field" as it is used in the book refers to:
a) The domain of the relation
plus
b) The converse domain of the relation
As I understand it the domain in this case is those members of a that can be mapped by P to one or members of b.
Similarly I understand the converse domain to be those members of b that are mapped from one or more members of a by P.
The bit that confuses me is the reference to the members of P.field. The way I understand it P.field is the map of a to b, produced by P.
In which case the only members of a in P.field are, by definition, mapped to members of b, and vice-versa for b.
So how can the minima of a include anything in P.field?
Perhaps my conception of "relation" isn't correct? The way I see it is as a map from the domain to the converse domain. e.g. A binary relation on a & b produces a list of a,b couples, a ternary relation of a's & b's to c's produces a list of a,b,c triples.
I'd be grateful if anyone can spot the cause/s of my confusion!