Re: Why not proper subset in definitions?

From: Adam (adam_at_bonkers.reg)
Date: 06/17/04 

Date: Thu, 17 Jun 2004 20:22:07 GMT

"Will Twentyman" <wtwentyman@read.my.sig> wrote in message
news:40d1bcba$1_4@newsfeed.slurp.net...
> > For instance, I thought of the following theorem,
> >
> > Theorem. If f: A -> B is a bijective function and S <= A, then
> > (i) The restriction of f to S is injective.
> > (ii) The restriction of f to S is not necessarily surjective.
>
> Part (ii) doesn't actually say anything.
>
> I'd rewrite it as (ii) if S < A, then the restriction of f to S is not
> surjective.

    Part (ii) does say something. It says that the restriction of f to S can
be surjective at times. The reason (ii) seems like it says nothing is
because of the definition of the restriction of f to S. Since I do not wish
to re-define terms, part (ii) would have been better worded as something
along the lines of "(ii) The restriction of f to S is surjective iff S = A,
as another poster wrote of. Since if S < A then the image of S under f|_S
can not equal B.

    Thanks, Adam.