Clarifying Notation

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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 :pac: - but it is a long post even
though it's dealing with just one idea :eek:

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 the .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,+',°)), •')
:eek: 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 :cool:

Fremen

I see you discovered Mathoverflow

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They didn't take too kindly to me, did they?

Fremen

Well, you got a sensible reply, and someone else had asked similar a question, so it's not the worst outcome in the world. You can't expect too much - it's a website for professional mathematicians to ask research questions (I've seen at least three fields medalists on there).

You can learn a lot from the "soft question", "intuition" and "big picture" tagged questions. Even if it's just finding new concepts to look up on wikipedia.

sponsoredwalk

Thats a great tip, those kind of sites rarely came up in all of my searches &
I only thought last night after I got those answers that them sites are really
worth specifically searching. Yeah I seen John Stillwell's name on that site
& knew it was a high level one but I thought this was at least a
graduate-level set theory question & not just my stupidity :P