FOM: ordered pair: Bourbaki

[Andrzej Trybulec]

On Sun, 2 May 1999 Vedasystem at aol.com wrote:

>  The Bourbaki's approach to introducing notation
> for ordered pair is preferable because one cannot prove
> silly theorems like (x,y) U {x} =  (x,y).

One cannot prove it anyway

    (x,y) U {x} = {{x},{x,y}} U {x} = {{x},{x,y},x} =/= (x,y)
by regularity.

You might think about an ordered pair as somthing consisting of
 unordered pair {x,y} and the order { (x,x), (x,y), (y,y) }
To avoid vicious circle, you may substitute in {x,y} the corresponding
lower sets. You get
     { {x}, {x,y} }
for triple (x,y,z)
  { {x}, {x,y}, {x,y,z} }

And for (x) 
 (x) = {{ x }}

 
Now:

    (x,y) U (x) = (x,y)

says that the union of a subset of a poset and the poset is the poset.
What's wrong about this.

Andrzej Trybulec