Re: what are keys and surrogates?

On 19 jan, 05:47, Keith H Duggar <dug..._at_alum.mit.edu> wrote:
> Jan Hidders wrote:
> > Keith H Duggar wrote:
> > > David BL wrote:
> > > > Jan Hidders wrote:
> > > > > In my experience as somebody who works in a Comp. science and
> > > > > Mathematics department the mathematicians that prefer the (D,C,G)
> > > > > definition of a function, also prefer the definition of relation that
> > > > > makes the domains explicit, and so define a binary relation as a
> > > > > triple (D_1,D_2,G). In that case a function is actually again a
> > > > > special case of a relation.  But this is all by no means
> > > > > uncontroversial. If you look at the entry for mathematical relation in
> > > > > Wikipedia you will see that there have been edit wars over this, and
> > > > > even one resulting in a ban.
>
> > > > LOL.
>
> > > > I agree that in a "single work" one should be consistent one way or
> > > > the other so that a function is indeed a relation.
>
> > > Finally you were able to admit "that a function is indeed
> > > a relation". Sadly you felt compelled to add the nonsense
> > > "single work" qualifier in a failed last-ditch attempt to
> > > save precise-boy face.
>
> > David is making a valid and correct point.
>
> No he wasn't. Perhaps you forgot too quickly that his original
> point (which he repeated in various ways) was:
>
>    "it [is] more precise to say that the
>     graph of a function is a relation"
>
> not that there are different definitions of "function".
>
> > There are many possible valid definitions of the notion of
> > function, even within mathematics, and it is not always the
> > case that functions are identified with their graphs.
>
> The only one to thus far claim that functions are associated
> with their graphs was David! Let's draw a simplified picture
> of the formalisms discussed and the various combinations one
> might choose
>
>    F1 = function is {(x,y)}
>    F2 = function is (D,C,G={(x,y)})
>    R1 = binary relation is {(x,y)}
>    R2 = binary relation is (D1,D2,G={(x,y)})
>
> possible combinations
>
>    (F1 R1) (F1 R2)
>    (F2 R1) (F2 R2)
>
> You yourself have pointed out under both (F1 R1) and (F2 R2)
> "a function is a relation". Furthermore, you also claim your
> experience indicates mathematicians choose either (F1 R1) or
> (F2 R2) not (F1 R2) nor (F2 R1) ie the one David holds high.
>
> Finally, who can know what David meant by "more precise" but
> I would choose (F2 R2) as more precise than his pet (F2 R1).
>
> Based on the above (mostly your own claims here simplified),
> would you not agree that "a function is a relation" is both
> more common and more precise than "the graph of a function
> is a relation"?

Before I answer that let me first agree that your simplified picture is correct. But as far as I understand him I don't think that David actually disagrees with that picture.

Do I think that the "a function is a relation" definition is both more common? Yes, I do. Do I think that it is more precise? The statement "the graph of a function is a relation" has the benefit of being true for several definitions of the notion of function. As such it might be preferable in the context of the c.d.t glossary where it is important to make clear what the different definitions are. Of course it is not really a definition but just the description of a certain property and in that sense less precise. But also here, I doubt that David would actually disagree with that.

• Jan Hidders