On Thu, 2010-12-16 at 20:22 +0900, Sebastian Fischer wrote:
> It is an equivalence class of pairs of (multi-)sets for the
> equivalence relation
>
> (A,X) ~ (B,Y) := A \union Y = X \union B
This was incorrect. The correct definition is (writing u for union)
(A,X) ~ (B,Y) := exists S : A u Y u S = X u B u S
As a consequence, if we use sets and not multisets, then every element
is equivalent to failure. Writing E for empty set:
(A,X) ~ (E,E) because for S = A u X : A u E u S = S = X u E u S
So, for this construction to be useful, one should consider
multiplicities of results such that (a ? a) is different from a.
We can also show that every element equals failure if (?) is idempotent
using the laws:
a = a ? failure
= a ? (a ? anti a)
= (a ? a) ? anti a
= a ? anti a
= failure
Sebastian
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Received on Do Dez 16 2010 - 15:39:13 CET