In article <_X4ek.4829$Ek.1538@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
> "Paul Sperry" <plsperry@[EMAIL PROTECTED]
> wrote in message
> news:110720082303066299%plsperry@[EMAIL PROTECTED]
> > In article <czRdk.60924$7v1.18433@[EMAIL PROTECTED]
>, Jack
<jj@[EMAIL PROTECTED]
>
> > wrote:
> >
> >> "Paul Sperry" <plsperry@[EMAIL PROTECTED]
> wrote in message
> >> news:110720081503413540%plsperry@[EMAIL PROTECTED]
> >> >
> >> >
> >> >
> >> > In article <8tMdk.2$Ek.1@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
[...]
> I can't say that I have a clear view of what the answer to my question
is. I
> am quite happy with the second of my definitions; it was written out for
me
> by a mathematician. The first, I lay down as 'Fix t to be the function
such
> that t: N --> N is arbitrary'; but doubtless this is inadequate...?
> I am just concerned about the other definitions that are dependent upon
t.
> Cheers.
>
>
OK. Here is your original post in this thread:
"I have got two definitions of t; the first I fix with the definition
let t : N ---> N be arbitrary'. In the second, I have t as the number
of primes in a set J that divide n. Immediately after the first
definition, I define c(n) as (t(n)(t(n)-1))/2. Is it OK to retain this
definition - without any statement - once I move on to my second
definition?"
I _think_ I know what you meant but let me deconstruct to see what you
actually _said_.
>I have got two definitions of t;
You already know this is going to get you in trouble.
>the first I fix with the definition 'let t : N ---> N be arbitrary'.
This isn't a definition and you can't "fix" an arbitrary function.
>In the second, I have t as the number of primes in a set J that
>divide n.
If you ignore what has gone before this is a perfectly good definition
of t : |N -> |N. At least relative to J - if you ever want two
different sets of primes in action at the same time you'll have
notational difficulties.
>Immediately after the first definition, I define c(n) as
>(t(n)(t(n)-1))/2.
This is a problem since your first "definition" didn't define anything
so c(n) could be virtually any number.
I think the first "definition" was in regard to the _letter_ "t" and it
was supposed to say that the letter "t" was going to stand for a
function from |N to |N.
Here is what I think you meant:
"Let J be a set of primes and for n in |N define
t(n) = |{p in J : p divides n}| and let c(n) = (1/2)*t(n)*(t(n) - 1)."
If this is indeed what you mean, you may now use "t" for the above
function until you explicitly change the meaning of "t". To elaborate
further, "t" is now the _name_ of the function that associates with
each non-negative integer the number of primes in J which divide it -
it is unwise to mistake the function for its name.
--
Paul Sperry
Columbia, SC (USA)
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