This question has been called "pedantic" and I agree, it is, but I doubt many people are able to answer it.

The basic problem is that I want to be extremely clear about the sets that mathematical manipulations and operations are taking place in, I find that an author referring to sets and functions just isn't enough when there exists a deeper structure that really clarifies concepts, as you'll see below when I mention linear algebra. Basically I want help exposing the ghost-like machinery in a clear and rigorous way.

The set-theoretic definition of a function is f = (X,Y,F) where F is a subset of ordered pairs of the Cartesian product of X & Y, (i.e. F ⊆ (X x Y) a relation). This is Bourbaki's way of defining a function and he (they) call F the graph.

But isn't a function itself a relation and therefore musn't we write (X,Y,f) as the set in which the function acts? To expand this out: (X,Y,f) = (X,Y,(X,Y,F)). I've come across notation that specifies (X,Y,f) as ((X,Y),f). Here, page 35 of this .pdf file So ((X,Y),f) = ((X,Y),((X,Y),F)) would seem to make sense.

Bourbaki calls f a set & F it's graph but the notation in the .pdf file says that f would be defined in the way I've explained above, i.e. that F is a subset of XxY. The thing is that since a function f is itself a relation shouldn't it be a relation in a set, i.e. ((X,Y),f)?

Assuming that the above is the way to think about these things, how would I think of both F & f? In f = (X,Y,F), F ⊆ (X x Y) so (x,y) ∈ F or xFy, where obviously (x∈X) & (y∈Y).

How about f? I think f ⊆ (X x Y) so (x,y) ∈ f or xfy.

I don't understand how this makes sense because for the set f = (X,Y,F) Bourbaki writes f : X → Y so for (X,Y,f) I'd have to set g = (X,Y,f) and write g : X → Y. This is a weird conclusion but it seems to suggest itself.

The problem of being extremely clear about what sets you are using is particularly interesting when doing linear algebra.

The use of set-theoretic notation in linear algebra both clarifies things for me and brings up similar questions, for a vector space V I could write ((V,+),(F,+',°),•) with the clarification that:

in (V,+) we have + : V × V → V,

in (F,+',°) we have (+' : F × F → F) & ( ° : F × F → F).

In • we have (• : F × V → V) or perhaps [• : (V,+) × (F,+',°) → (V,+)]?

This notation clearly illustrates why the two operations, vector addition and scalar multiplication are used on a vector space and the axioms for each clearly jump out, i.e. (V,+) is abelian, (F,+',°) is a field and • isn't the clearest to me but I think it's similar to the way that + & ° are related in a field, i.e. "multiplication distributes over addition".

Relating all of this to the concerns I had above in a clear manner, in the set (F,+',°) it would make sense that +' is a set of the form (F,+'') where +'' is a subset of the cartesian product of F x F. Similarly with °, and in the set (V,+) you'd have something similar, also in • you'd have a crazy set ((V,+), (F,+',°), •') or including even more brackets (((V,+), (F,+',°)), •') with •' being a subset of the cartesian product of (V,+) & (F,+',°).

There is another problem when you want to give a vector space a norm, would I write ((V,+),(F,+',°),•,⊗) where ⊗ : V x V → F ? Would ⊗ itself suggest the subset ((V,+), (F,+',°), ⊗') in the manner explained above? I don't think so because ⊗' would be the set of ordered pairs (x,a) with x ∈ V and a ∈ F but since V x V → F you've got the map (x,x') ↦ a, it's quite confusing tbh and need help with this.

All this seems crazy but it also makes a lot of sense, I want to be very rigorous about what I'm doing and all of the above seems to suggest itself but it could be a lot of nonsense caused by simple confusion of a particular issue in the post , I'm thinking that (X,Y,F) implying (X,Y,f) is the culprit but again this idea clarifies things. If you read to this point thanks so much, hopefully you recognise the issue.