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Re: RM and definition of relations/tuples

From: Jonathan Leffler <jleffler_at_earthlink.net>
Date: Sat, 26 Nov 2005 07:14:07 GMT
Message-ID: <3XThf.1293$Hk1.579_at_newsread1.news.pas.earthlink.net> 






Martin Zapf wrote:

> vc wrote:
> 




>> Martin Zapf wrote:


>>

>>> I have a question to the Relational Model and the definition of

>>> relations and tuples.

>>>

>>>

>>> I learned the following definitions:

>>>

>>> A relation schema R is a set of Attributes R={A_1,A_2,...,A_n}

>>> Each Attribute A has a domain dom(A)

>>>

>>> Here comes the problem, there are two definitions for relations/tuples

>>>

>>> 1. defintion:

>>>

>>>     A relation r for schema R is a mathematical relation (cartesian

>>>     product) over the

>>>     domains from the attributes of R.

>>>     So r:=dom(A_1)xdom(A_2)x...xdom(A_n)

>>>     A tuple is an element of r.

>>>

>>> 2. definition

>>>

>>>     A relation r for schem R is a set of tuples.

>>>     A tuple t is a function

>>>     t: R -> Union (dom(A_1),dom(A_2),...,dom(A_1))

>>>     t maps each Attribute of R to an value of its domain.

>>>     So a relation is a set of functions, which are called tuples.

>>>

>>> I noticed, that the difference between this two definition is that

>>> definition 1 forces a certain order for the Attributes and the values

>>> for them in the tuples.

>>> The 2. definition doesnt need any order for the attributes.

>>>

>>> Im quite confussed, is there a "better" definition or should you always

>>> use both?

>>

>> The first definition is closer to the "mathematical" relation where a

>> tuple is an *ordered* sequence.  The second definition is what database

>> folks prefer to use (see Codd, Date, et al) where a tuple is a *set* of

>> attribbute:value pairs.  One can be mapped to the other and I would not

>> worry too much about the difference unless you are doing some d.b.

>> theoretical research.

> 
> I also figured that out, a mathematical relations (subset of catesian 
> product) is ordered because the elements of a cartesian products are 
> ordered sets by definition.





I don't think so.  The mathematical relation is an unordered set of 
ordered tuples.  The cartesian product is a unordered set too - of 
ordered tuples to be sure.



I think of an ordered tuple as one where the elements are implicitly 
named by index number (position in the tuple); in comparison, a tuple in 
the relational model has explicitly named elements, and therefore they 
form a set of elements (one reason you can't have several attributes 
with the same name) and do not need to be explicitly ordered.  I find 
this analogy/contrast helpful; you may or may not find it so.


> And yes I have to do some theoretical db stuff so Im asking myself:
> If there are two definitions for the same thing, what is the universal 
> valid and precise definition for the RM?  This two definitions cant come 
> out of nowhere, can they.





There are many definitions for many things.  "Infinite are the arguments 
of mages" - a (mis?)quote from a sci-fi story.  Similarly, different 
people in the DB world have different versions of the definition.  There 
isn't any one universal truth.


-- 
Jonathan Leffler                   #include <disclaimer.h>
Email: jleffler_at_earthlink.net, jleffler_at_us.ibm.com
Guardian of DBD::Informix v2005.02 -- http://dbi.perl.org/


Received on Sat Nov 26 2005 - 08:14:07 CET












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