Re: Why is the cartesian product...

In article <1130034083.778853.38630@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<matthewpanas@xxxxxxxxx> wrote:
>It's generally accepted that the cartesian product of a set A and the
>empty set is the empty set, but why? Is it because we define the
>cartesian product A x B of sets A and B as (a,b) where a is in A and b
>is in B, and since A x (empty set) is not defined, it is the empty set?

Essentially. One is usually a bit more precise, depending on how you
define your ordered pairs; under the usual definition of (a,b) as
{{a},{a,b}}, then the cartesian product is defined as a particular
subset of P(P(AUB)), and in the case when B is empty, one can show
that the subgroup thus defined is empty.

An alternative is to define the cartesian product of A and B as the
set of all maps from the set {1,2} into the set A U B, such that f(1)
is in A and f(2) is in B. Since there are no functions that satisfy
the conditions, the set of all such maps is empty.


>I would think this would be not defined...

The set is perfectly well defined. The fact that it does not have
elements does not make it undefined.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

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