Re: Do Data Models Need to built on a Mathematical Concept?

From: Neo <neo55592_at_hotmail.com>
Date: 5 May 2003 08:43:15 -0700
Message-ID: <4b45d3ad.0305050743.30acb4de@posting.google.com>

> > Pg 65 of Schaum's Outline (0-07-038159-3), binary relation as follows:
> > A = (b,c)
> > B = (j,k,l)
> > AxB = ((b,j), (b,k), (b,l), (c,j), (c,k), (c,l))
> > then (b,j), (b,k) ... are binary relations.
>
> That's not the generally accepted definition of the term. Nor is it even
> what that specific book says; I checked. What is says is:
> "Let A and B be sets. A *binary relation* or, simple,
> a *relation* from A to B is a subset of A x B."
> So, to fix your example, the cardinality-2 set { (b,j), (b,k) } is
> a binary relation.

Ok, let me try to understand:
"A *binary relation* from A to B is a subset of A x B."

Since (NULL) is a subset of AxB, it is a binary relation!!!
Since ((b,j)) is a subset of AxB, it is a binary relation.
Since ((b,j), (b,k)) is a subset of AxB, it is a binary relation.
Since ((b,j), (b,k), (b,l)) is a subset of AxB, it is a binary

relation.
and so on....

If the above is the correct interpretation, I find the definition to be inaccurate in that it has not defined the fundamental form of a binary relation.

In any case (b,j) is not A binary relation, it is composed OF two binary relations: b is directly related to A, and j is directly related to A, therfore, b and j are indirectly related but via A.

> To expand on the comparison a little:
> An example of a binary relation on Z x Z (the integers) is 'less than'.
> So the set of all integers a, b where a < b is a binary relation. If your
> explanation of the standard definition were correct, (which I do
> understand you specifically disagree with) then we would say
> '1<2' and '7<123' are binary relations.

I would say '1<2' is not A binary relation but composed of TWO direct binary relations. Most people only see the relationship between 1 and 2. I see it as 1 is related to its set(ie whole numbers) AND 2 is related to its set(ie whole numbers). The < is an indirect or derived relationship between 1 and 2.

However, I think I see where my perceptions mismatches or errors from that of the rest. The rest see three nodes and TWO links as a binary relation, the rest count relations. I have been calling TWO nodes and one link a binary relation, I counted things. Is this where my mistake lies?

Set theory books say X-Y-Z is a binary relation because there are two link.
I have been saying X-Y is a binary relation because there are two nodes.

But now my symbolic logic books say "If it connects two elements at a time it is said to be a dyadic relation, if three, triadic". Here they count things.

As best as I can determine, the word "binary" before the word "relation" could be interpretted in two way: 1. indicates the number of relations.
2. a relation between 2 things. Received on Mon May 05 2003 - 10:43:15 CDT