expanding cross products

[todd]

How do you expand a cross product like ((A × B) × C) using the
definition of the cross product? I would know how to do it if it were
just (A x B).

[Ian W.]

(A x B) is a set, and C is a set. With that in mind:

((A x B) x C) = {<x,y> : x in (A x B) & y in C}.

The next step is to expand (A x B) in the formula inside the set-
builder notation.


It might also be helpful to remember that when we write:

{<x,y> : phi},

this is just shorthand for:

{z : z = <x,y> & phi}.

Ian

[dsny...@gmail.com]

So what is the cross product of two pairs?

Such as (1 . 2) x (3 . 4)?

[David Rager]

( (1 . 2) . (3 . 4) )

[Anar Seyf]

It would have made more sense if 358.1 and 358.2 were named 371.1 and
371.2. In fact answering 371 provides sort of a hint to .1 and .2.

On May 5, 12:30 pm, David Rager <rage...@gmail.com> wrote:
> ( (1 . 2) . (3 . 4) )
>

> On Tue, May 5, 2009 at 12:25 PM, dsnyde...@gmail.com

[Ian W.]

This seems unclear, because I think the cross product is only defined
on sets. David's answer seems to assume that your sets are represented
as lists, but (1 . 2) is not a true list. So I would say that (1 . 2)
x (3 . 4) is actually not defined.

Ian

On May 5, 12:30 pm, David Rager <rage...@gmail.com> wrote:

> ( (1 . 2) . (3 . 4) )
>

> On Tue, May 5, 2009 at 12:25 PM, dsnyde...@gmail.com

[David Rager]

Indeed, the correct question would concern, in ACL2 notation:

( (1 . 2) ) x ( (3 . 4) )

And the answer would be

( ( (1 . 2) . (3 . 4) ) )


In roster notation the question would concern:

{ <1, 2> } x { <3, 4> }

And the answer would be:

{ < <1, 2>, <3, 4>> }