Re: Review of Mueckenheims book.

"MoeBlee" <jazzmobe@xxxxxxxxxxx> writes:

While the mapping from omega to its power set in which each element is
mapped to itself, is an injective mapping, the term "identity" when
applied to mappings usually requires a bijection with the codomain
identical to the domain.

By 'codomain', I take it that you mean the range.

I don't think so. He means codomain.

In category theory, a function is not just a set of ordered pairs. A
function requires that we specify the source and target. So, for
instance, we write id_N:N -> N. The function you're talking about is
formed by taking the inclusion N -> P(N), but this inclusion is a
different function than id.

Of course, in set theory, a function is just a set of ordered pairs
and so there is no need to pick out a domain and codomain. Both id_N
and the inclusion are, in this sense, the same function.

But Virgil is think in category-theoretic terms. (I doubt that
category theory explains Mueckenheim's confusion however.)

--
Jesse F. Hughes

"How lucky we are to be able to hear how miserable Willie Nelson could
imagine himself to be." -- Ken Tucker on Fresh Air
.

Relevant Pages

  • Re: Review of Mueckenheims book.
    ... By 'codomain', I take it that you mean the range. ... a function is not just a set of ordered pairs. ... formed by taking the inclusion N -> P, ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... By 'codomain', I take it that you mean the range. ... formed by taking the inclusion N -> P, ... In pure set theory, perhaps, and so pure set theory is almost useless. ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... applied to mappings usually requires a bijection with the codomain ... He means codomain. ... formed by taking the inclusion N -> P, ...
    (sci.math)
  • Re: Why f:domain->codomain instead of f:domain->range?
    ... >>> when it is much easier to describe a function's codomain than than ... Or are there more fundamental reasons ... The official definition of a function as a set of ordered pairs ... The categorical usage, on the other hand, has many things to recommend ...
    (sci.math)
  • Re: Question Regarding the Definition of Cantors Set
    ... My mistake - should have written the codomain of f is P. ... In this case, if a function was just a set of ordered pairs, then it ... be made referencing elements at a higher level. ... Is this understanding incorrect? ...
    (sci.logic)
Quantcast