From rapaport@cse.Buffalo.EDU  Fri Apr 18 07:13:41 2008
Date: Fri, 18 Apr 2008 07:13:41 -0400 (EDT)
From: "William J. Rapaport" <rapaport@cse.Buffalo.EDU>
To: CSE191-SP08-LIST@LISTSERV.BUFFALO.EDU, rapaport@cse.Buffalo.EDU
Subject: Re: 191-Transitivity

Oops!

> ------------------------------------------------------------------------
> Subject: 191-Transitivity---CORRECTION CORRECTION CORRECTION!!!!!!!
> ------------------------------------------------------------------------
> 
> A student asks:
> 
> > I had a question about transitivity.
> > Is the following relation transitive?: R={(a,b) , (b,b)}
> > Because here we can go from a to a and from b to b and hence we can go from a to
> > b. ...
> 
> Yes, it is.  To be transitive, a relation has to satisfy:
> 
> (for all x,y,z)[R(x,y) ^ R(y,z) -> R(x,z)]
> 
> There are only 2 possibilities for the antecedent:

NOPE!  There's only ONE possibility for the antecedent that reflects the
relation R, namely:

R(a,b) ^ R(b,b)	[it doesn't matter which order I write it in;
		"^" is commutative!]

And that does indeed imply R(a,b), so R is transitive.

Any other "R" statement, like "R(b,a)" or "R(a,a)" is false, and if any
of *them* were in the antecedent, THEN the conditional would be
vacuously true.

As I correctly said in my previous posting, R *is* indeed transitive.