- #1

- 532

- 5

**This post is an example of the danger of confusing notation, no need to read on!**

I was confusing Bourbaki notation with set notation and mixing them

together in a really messy way, drove me crazy!

The basic problem is that I want to be extremely clear about the sets that

mathematical manipulations and operations are taking place in, I am hoping for

someone who really understands this to read what I've written closely and point

out what is getting me all mixed up, though of course reading &/or responding

isn't mandatory :tongue2: - but it is a long post even though it's dealing with just

one idea :yuck:

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).

I copied this out of Bourbaki's set theory.

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).

*http://www.math.bilgi.edu.tr/courses/Lecture%20Notes/SetTheoryLectureNotes.pdf [Broken]*

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 itself, 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.

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

I don't understand how this makes sense because in 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.

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,+',°)), •')

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, 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 , thanks for taking the time

Last edited by a moderator: