[FOM] A big trouble with the definition of "binary relation"

Arnon Avron aa at tau.ac.il
Thu Feb 1 17:17:08 EST 2007

On Sun, Jan 28, 2007 at 11:43:14AM -0500, Victor Makarov wrote:
> The following definition of "binary relation" one can find in 
> Wikipedia:
> ( http://en.wikipedia.org/wiki/Binary_relation#Formal_definition  )
> "A binary relation R is usually defined as an ordered triple (X, Y, 
> G) where X and Y are arbitrary sets (or classes), and G is a subset 
> of the Cartesian product X ? Y."

Without entering the concrete question posed in the message, I would
like to note that modern textbooks in set theory or logic do not
define a binary relation R as a triple, but simply as a set of
ordered pairs. R is then a relation from X to Y iff it
is a a subset of the Cartesian product of X and Y. Indeed, there
seems to be no sense in taking the exponential function from
the reals to the reals and the exponential function from
the reals to the positive reals as two different objects.

  The real problem with these definitions of a binary relation
is that some of the most basic things called "relations"
in mathematics are not relations at all according to them.
Thus the two basic relations of set theory, the epsilon realtion
and equality, are obviously not sets, or even classes, of ordered
pairs. In fact one needs to understand these two "relations" well
before one can even start to talk about relations as subsets of
cartesian products (the \subset "relation" is another thing
which is not a relation according to the official definition
of this concept - yet we all teach our students that it is
a transitive "relation").

 I am having hard time every year in explaining this subtle
point to students. I wonder what others do about it (the easiest
solution is of course not to mention the problem at all, unless
some student asks about it -  which almost never happens...).

Arnon Avron

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