Re: General Form of Relationships?

In article <4b45d3ad.0212281437.3102c5f9_at_posting.google.com>, Neo <neo55592_at_hotmail.com> wrote:
>What is the general form of relationships? Is it
>1) "t1 relator t2" or
>2) "relator, t1, t2, ..."

The most general way to view a "relationship" among a set of n variables is as an arbitrary subset of n-space. (In your examples, the variables may come from other sets S_1, ..., S_n instead of the real number line, in which case a "relationship" is a subset of the cartesian product S_1 x S_2 x ... x S_n .) For example, the relationship "<" can be described by handing someone the set of all pairs (x,y) with x<y ; this is an open half-plane.

>Can form 2 relationships always be broken down to multiple form 1's?
>Or are some relationships such that they cannot be broken down
>and thus form 2 is the more general.

You'll have to describe what "broken down to" means. In some sense every relationship can be described using "and", "or", and relationships of one variable, that is, every subset can be described using unions, intersections, and projections to a single coordinate: Write the set as a union of points specified by each of their coordinates. Of course this may require an infinite number of "or"s (unions of singletons).

Mathematically the question gets more interesting if you restrict to relations which are graphs of continuous functions: Hilbert's 13th problem asked whether continuous functions of several variables can always be reduced to functions of one variable, using addition and composition. (There is an affirmative answer; see e.g. http://www.math-atlas.org/98/hilb_13) I don't know if there is a good extension to relations other than (graphs of) functions.

Follow-ups set to sci.math.

dave Received on Mon Dec 30 2002 - 03:42:56 CET