Re: Part 2 WD error?: inverse of >: < or <=?

At 6:13 PM -0400 2006-06-19, Daniel Barclay wrote:
>In the XML Schema 1.1 Part 2 working draft currently at
>http://www.w3.org/TR/2006/WD-xmlschema11-2-20060217/datatypes.diff-1.0.html,
>section 2.2.3 says:
>
>     In this specification, this less-than order relation is denoted by '<'
>     (and its inverse by '>'), the weak order by '¾' (and its inverse by
>     '„'), ...
>
>Is that backwards?  "not ( x > y )" does not imply "x < y"; it implies
>"x ¾ y".
>
>In which sense is "inverse" meant?  Is "inverse" the right word?  Should
>it be "converse"?  something else?

I suspect you've confused "inverse" with "negation".  The inverse of a
relation is never the same as its negation.

Think of the relation as a set of ordered pairs.  "Inverse" then means
"reverse each pair".  I.e.: in general, x R y iff y invR x; in particular
here, x < y iff y > x.

<asides type="more than you ever wanted to know, probably">

This is "relation" terminology.  The word "inverse" is also used WRT
commutative binary operations that have an identity.  E.g., negation
is the in inverse WRT addition; "reciprocation" is the inverse WRT
multiplication (at least for real numbers).  In this usage, if you
replace "WRT" by "of", you still don't get negation.  The only place
where this kind of inverse is negation is if the operations are those
of Boolean algebra.

It does turn out that the relational inverse of a one-to-one function
is that function's binary-operation inverse WRT composition of functions.
(Providing that you take the view that a function is a special kind of
relation.  Not everyone does.)

Furthermore, in logic when dealing with the implication relation one
generally uses the word "converse" for this reverse relation; "inverse"
is then used for the contrapositive of the converse.  Not much confusion
is engendered thereby since it is a theorem of logic that an implication
is always equivalent to its contrapositive, hence logical "inverse" is
always equivalent to "converse", which is the relational "inverse".  ;-)

It might have made sense for set theorists to follow the terminological
lead of the logicians, but it appears they didn't.  One rarely sees
"converse" used for relations other than logical implication.

</asides>

Hope this helps.
-- 
Dave Peterson
SGMLWorks!

davep@iit.edu

Received on Tuesday, 20 June 2006 04:19:10 UTC