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Replies: 23   Last Post: Nov 29, 2012 1:13 AM

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Dan Christensen

Posts: 6,588
Registered: 7/9/08
Posted: Nov 28, 2012 10:13 AM
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On Nov 28, 1:35 am, Graham Cooper <> wrote:
> On Nov 28, 2:26 pm, Dan Christensen <>
> wrote:

> > On Nov 27, 8:59 pm, Graham Cooper <> wrote:
> > > There are 2 ALLs which is more complicated but you can format it as a
> > > SUBSET using a cartesian product of the 2 X values with a common Y.

> > > isfunction(r)  <-  ALL(Y1) ALL(Y2) r(X,Y1)^r(X,Y2) -> Y1=Y2
> > There is more to functionality than this. I may not fully understand
> > your unusual notation (PROLOG?), but it would seem you have left out
> > the requirement that FOR ALL elements of some domain set, THERE EXISTS
> > a unique image in a codomain set. (This is where I think quantifiers
> > become indispensable).

> > > {(Y1,Y2)|r(X,Y1)^r(X,Y2)} C {(Y1,Y2)|r(X,Y1)^r(X,Y2)->Y1=Y2}
> > [snip]
> > The same comment applies... I think.
> > Anyway, I am still waiting for proofs of the following:
> > 1. {(x,y) | x in S, y=x}  is a function mapping the set S onto itself
> > 2. {x | ~x in x} cannot exist
> > 3. {x | x=x} cannot exist

> > You really need to address these fundamental results.
> > For what it is worth, and from what little I know about PROLOG, it
> > doesn't seem to be capable of all that is required to do mathematical
> > proofs in general. It may be able to model some interesting and useful
> > aspects of predicate logic and set theory, but, for your purposes,
> > important pieces of the puzzle seem to be missing.

> Nope, this is exactly the definition of function.
> {(Y1,Y2) | r(X,Y1)^r(X,Y2)}  C  {(Y,Y) | r(X,Y)}
> which simply guarantees only 1 Y value for any X value.

This is a common mistake. According to this erroneous view, every set
{(x,y)} is a function for any objects x and y. The functionality of a
set of ordered pairs is always defined in terms of a domain and
codomain set. Here is a typical formal(ish) definition of a function
from Wiki (my comments in []'s):

"A function f from X [the domain of f] to Y [the codomain of f] is a
subset of the Cartesian product X × Y subject to the following
condition: every element of X is the first component of one and only
one ordered pair in the subset.[3] In other words, for every x in X
there is exactly one element y such that the ordered pair (x, y) is
contained in the subset defining the function f."

Example: Let X=Y={0,1}.

Then {(0,0), (1,1)} is function from X to Y, while {(0,1)} is not.

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