Re: Review of Mueckenheims book.

In article <1173826889.935164.189530@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:

On Mar 13, 3:05 pm, Virgil <vir...@xxxxxxxxxxx> wrote:

There are all sorts of important properties of relations which can only
be determined when their domains and codomains are explicitly specified.

Correct, though, it would be better to say 'domain' and 'supersets of
the range'.

AS in your mind, the two are congruent notions, why is the longer and
more awkard any better?

And nothing I said contradicts that in many instances it
is crucial that we discuss, say, a particular superset of the range.

In fact, I said at least a few times that there are such instances.
Even in proof of the definition by recursion theorem in Halmos (or
Enderton or Suppes or whatever), it crucial to prove that the function
is into a certain set.

Absent them, how does one tell if such a relation
is an equivalence relation?

By proving or disproving that it satisfies the definition of an
equivalence relation.

But whether a set of ordered pairs is an equivalence relation depends on
the cartesian product of which it is a subset, so it cannot be proven
either way without that being specified.

That is in such common textbooks as Enderton and Suppes that I am
surprised you ask.

Or an order relation?

By proving or disproving that it satisfies the definition of a
whatever kind of ordering we are asking about.

That is in such common textbooks as Enderton and Suppes that I am
surprised you ask.

And when that definition requires a specific domain or codomain, as it
often does?

Or an injective relation?

By proving or disproving that it the function satisfies the definition
of an injection.

A relation can be injective without being a function.

And as to whether the function is an injection into a
mentioned set, by proving or disproving that the function and set
satisfy the definition of '___ is an injection into ___'.

I was specifically asking about relations, which need not always be
functions.

That is in such common textbooks as Enderton and Suppes that I am
surprised you ask.



Or a surjective relation?

By proving or disproving that it the function satisfies the definition
of a surjection. And as to whether the function is an surjection onto
a mentioned set, by proving or disproving that the function and set
satisfy the definition of '___ is surjection onto ___'.

As it need not be a function at all to be a surjective relation, your
answer is irrelevant

That is in such common textbooks as Enderton and Suppes that I am
surprised you ask.

As it need not be a function at all to be a surjective relation, your
comment is irrelevant

Or a symmetric relation?

By proving or disproving that the relation satisfies the definition of
a symmetric relation.

Which cannot be done without knowing the domain and codomain.

That is in such common textbooks as Enderton and Suppes that I am
surprised you ask.

Then you had best reread those texts with more case yourself.

Or any of many other things dependent on a known domain and codomain.-

If it is desired to prove something about a relation with respect to
its domain and/or any given superset of its range, then, in set
theory, we apply first order logic to the axioms and definitions, as
we always do.

And since many such depend specifically on both domain and codomain, one
cannot answer such questions on the basis of a set of ordered pairs
alone.
.

Relevant Pages

  • DISPROVING07, 2nd CfP
    ... Automated Reasoning traditionally has focused on proving ... proving, which we call disproving, particularly aims at identifying ... CADE 2005, and FLoC 2006. ... Authors of papers presented at DISPROVING and VERIFY will ...
    (comp.specification.z)
  • DISPROVING07 - DEADLINE EXTENDED to May 25, 2007
    ... Last Call for Papers ... Automated Reasoning traditionally has focused on proving ... proving, which we call disproving, particularly aims at identifying ... CADE 2005, and FLoC 2006. ...
    (comp.specification.z)
  • DISPROVING 06 2nd Call f. Papers
    ... The opposite of proving, which we call disproving, particularly ... aims at identifying non-theorems, i.e. showing non-validity ... logics like for instance program logics. ...
    (comp.specification.z)
  • FLoC06 WS on DISPROVING Call for Papers
    ... The opposite of proving, which we call disproving, particularly ... aims at identifying non-theorems, i.e. showing non-validity ... logics like for instance program logics. ...
    (comp.specification.z)
  • Re: Injectuve and Surjective function difficulty
    ... >> make the codomain of t R, you will never get a negative value. ... > method for proving that something is not surjective. ... To prove something is not surjective is simply a matter of exhibiting a ...
    (sci.math)