Re: RM and definition of relations/tuples

Jonathan Leffler wrote:
> 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.

You are right, I did describe that wrong. The cartesiaon product or relation itself is of course not ordered. Only the components inside their elements are ordered.

> 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.