Paul,
>> Paul,
>>
>> > Let |N be the set of non-negative integers.
>> > Let I be a finite interval of integers.
>> >
>> > DEFINITION 1. For a function t : |N -> N define
>> > c_t(n) = (1/2)*t(n)*(t(n) - 1).
>>
>> Will I need to say anything about the integer n? Or even the standard
|N?
>> Or
>> can I take their meanings as understood?
>
> It never hurts to nail things down. If |N is not explicitly given there
> may be a question of whether or not 0 is included.
>
> It certainly wouldn't be amiss to make the definition explicit:
>
> DEFINITION 1.1 For a function t : |N -> N define a function
> c_t : |N -> |N by c_t(n) = (1/2)*t(n)*(t(n) - 1).
I'm just worried that I'm using the lower case 'n' without any prior
definition.
Incidentally, where you give the second definition,
<< DEFINITION 2. For a set of primes J define t_J : |N -> |N by
t_J(n) = |{p in J : p divides n}|>>
I wonder what you think of this. I have defined J as a subset of a set of
the first m primes, where m is an integer, which I denote by P(m). In
accordance with your advice, I reference the function t_J in the notation
(x,y,t_J) without comment. Now, if I want to refer to a set J comprised of
all m primes, I make the reference (x,y,t_P(m)), again without comment. Is
that acceptable?
>> I got this further issue. Brian set me up with a definition which goes
>> 'We
>> will find it convenient to define auxilliary functions in the following
>> way:
>> given (g,n) in G there is a unique (s,r) in G', such that h(g,n)=(s,r),
>> and
>> we set h_1(g,n)=s and h_2(g,n)=r'. Can I take this to be a definition
>> that
>> holds in my reference to s and r subsequently, or do I have to go
through
>> the process of saying what they are every time I make a reference to
>> them?
>> If so, what about my later use of r to denote memebrs of R?
>
> It would appear that you are free to recycle "r" and "s".
>
I take it, then, that if r is in R, it is invariably tagged such that it
is
referenced as r \in r. If I don't tag r as such, will the reader know that
I
mean it to be read in its other sense, i.e. as h_2(s,r)?
>> More to the
>> point, if I define j(x,y,t) to denote a specific thing, can I casually
>> say,
>> later on, "take any integers i and j", without any reference to
j(x,y,t)
>> being meant?
>
> In general, no. I have no context but "j" cannot be simultaneously the
> name of a function and the name of an integer.
Ah, trouble. Will run out of letters that way. :?(
With thanks.
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