[formal-methods] Injective, Surjective, and Bijective functions

[Crutcher Dunnavant]

I find un-cited definitions un-compelling evidence.

I'm perfectly willing to listen to argument, and read proofs. But for
appeals to authority, I would like citation.

As to the issue at hand, it seems trivial to say that injection abstracts to
monomorphism and surjection abstracts to epimorphism. Awody's book (Category
Theory) makes this claim on page 25. What I haven't seen anywhere, and I've
looked, is a claim that it is incorrect to say that injection and surjection
are synonyms for their more general concepts.

Jason suggests that this is merely a cultural or political distinction.

On Jun 27, 2009, at 4:05 PM, d p chang wrote:
>
> > Mikael Vejdemo-Johansson <mik at stanford.edu> writes:
> >
> >> On Jun 27, 2009, at 12:11 AM, d p chang wrote:
> >>> Mikael Vejdemo-Johansson <mik at stanford.edu> writes:
> >>>> For instance, the map (*2): Z -> Z is injective (prove it!) but not
> >>>> surjective (prove it!), and is an establishing map for why
> >>>> infinities
> >>>> are weird.
> >>>
> >>> i'm not sure i have the 'notation' for reasoning/expressing that the
> >>> infinities are equal, but it seems like one of those questions i
> >>> associate w/ cantor and diagonals.
> >>
> >> Define |S| <= |T| iff there exists an injection S -> T.
> >>
> >> Also, define |S| = |T| iff there exists a bijection S -> T.
> >>
> >> But still, we can fit |Z| elements within |Z| elements and still get
> >> gaps. This cannot happen for the finite case - any strict injection
> >> S -
> >> T (not a bijection) of finite sets is a proof that |S| < |T|, but
> >> it breaks for infinite sets.
> >
> > thanks for that.
> >
> > i'm still thinking about how (*2) over Z is injective. it seems like i
> > have to reason over the function itself.
> >
>
> f is injective iff f(x) = f(y) implies x = y.
>
> Suppose 2n = 2m for n,m in Z. Can we conclude that n=m?
>
> Mikael Vejdemo-Johansson, Dr.rer.nat
> Postdoctoral researcher
> mik at math.stanford.edu
>
>
>
>
>
>
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-- 
Crutcher Dunnavant <crutcher at gmail.com>
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