Re: curiousity of sets of no relations?
Date: Sat, 10 Jun 2006 18:35:15 GMT
paul c wrote:
>> paul c wrote: >> >>> ... >>> That seems a practical motivation. In terms of relations and/or set >>> theory/predicate calculus can anybody give a more theoretical one? >> >> He simply defined them as the identity elements for the specific >> operations just as one defines any aggregate/fold of zero elements >> using the identity element for the base operation.
> In other words, I'm guessing, "to make it all work/to guarantee the ops
> always return relations", i.e., the motivation is practical only,
> analogous to notions such as the "empty product" (such as in
> http://en.wikipedia.org/wiki/Empty_product)? If so, I think I'm content
> with that, e.g., thinking of the empty set as just a gizmo to enable
> various operations we desire.
Exactly so. And exactly the same way Graham, Knuth and Patashnik define aggregation in _Concrete Mathematics_. We define the iteration over zero items as the identity element of the base operation. Sum() = 0, Product() = 1, Min() = max_value_of_type, Max() = min_value_of_type, etc.
Mathematicians define aggregations that way on utilitarian grounds. Received on Sat Jun 10 2006 - 20:35:15 CEST