Re: ordered pair
From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 08/30/04
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Date: Mon, 30 Aug 2004 18:42:48 +0000 (UTC)
In article <Pine.GSO.4.58.0408301054150.27670@ascc.artsci.wustl.edu>,
Michael Hamm <msh210@math.wustl.edu> wrote:
>Is there a standard way of defining ordered pair (or n-tuple)? If so,
>what is it? (I've seen (x,y) defined as {{x},{x,y}} iIrc; is that
>standard?)
It is the most common way of defining it in Set Theory. Once you have
ordered pairs, you can define functions. Once you have functions, the
"best" way to define an n-tuple of elements of A is to define it as a
function whose domain is the natural number n = {0,1,2,3,...,n-1}, and
whose codomain is A. Then the i-th "entry" is the value f(i). This
allows you to define "ordered tuples" for any ordinal, not just the
finite ones.
You ->could<- define ordered n-tuples "directly" the same way as you
do ordered pairs: the tuple (x_1,...,x_n) could be defined to be the
set { {x_1}, {x_1,x_2}, {x_1,x_2,x_3}, ...., {x_1,x_2,...,x_n} }; but
the functional point of view is more typical.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu
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