Re: bijective mapping of sets

In article <1143161498.846509.112430@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<sikari.h@xxxxxxxx> wrote:

Is this reasoning generally valid to show that
two sets are equal and also that f is injective;
I mean is the following true:
\forall A,B \subset X such that
A \subset B and f(X-A)=f(X-B) => B \subset A (i.e. B=A) and f is
injective

You are not being very careful with your notation; specifically, it is
not clear what the scope of the quantification is, and whether the
implication includes the "and" or not. If you mean:

If f:X->Y if a function, such that
for all subsets A and B of X, if A is contained in B and
f(X-A) = f(X-B) then B is contained in A
then f is injective

then the answer is yes: given any a,b in X such that f(a)=f(b), then
letting A={a}, B={a,b}, we have A contained in B, f(X-A)=f(X-B), so we
conclude that A=B, and hence that a=b, which proves f is injective.

On the other hand, you could mean:

Let f:X->Y be a function. Then for all subsets A and B of X, [if A is
contained in B and f(X-A)=f(X-B), then A=B]; in particular f is
injective.

Then this is not true: a constant function will f(X-A)=f(X-B) in all
cases except when one of A and B is all of X and the other one is
not.




--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

Relevant Pages

  • Re: make 100 by using 1, 7, 7, 7, 7
    ... The implication of showing four 7's is that you are supposed to use ... what I accept as reality." ... Arturo Magidin ...
    (sci.math)
  • Re: Models. predictions, physicallity, and observations
    ... You've defined reality as observations, ... >> changing simply points to why any discussion of reality has no place in physics. ... >> quantification, and you know that quantification is an intellectual process. ...
    (sci.physics)
  • Re: Models. predictions, physicallity, and observations
    ... You've defined reality as observations, ... >> changing simply points to why any discussion of reality has no place in physics. ... >> quantification, and you know that quantification is an intellectual process. ...
    (sci.astro)
  • Re: Models. predictions, physicallity, and observations
    ... You've defined reality as observations, ... >> changing simply points to why any discussion of reality has no place in physics. ... >> quantification, and you know that quantification is an intellectual process. ...
    (sci.physics.relativity)
  • Re: Uncertainty and Continuity?
    ... |> It seems that if someone says that reality is ultimately unknowable, ... |> quantification cannot be expressed by a finite string of symbols. ... That I like it and that the quantification via mathematics has the same ...
    (sci.physics.relativity)
Quantcast