Let's say we have a set $B$. Some of its elements can have two versions, let's say $b^+$ and $b^-$, while the rest can only have one: $b^+$.
Now I want to build a set that contains the two versions of those elements that can have them, and the only one version for those elements which have only one.
So my question is: How do I build "derived" sets ? Which formalizes:
- $X$: The set of "$^+$" versions of all elements of $B$
- $Y$: The set of "$^-$" versions of the elements of $B$ which can have them (assume there is a predicate that can be used to detect them)
- Implies $X\cup Y \neq B$
Example:
$B$ can be a set of cars, some of which come in 4wd or 2wd versions, and some only come in 2wd. I'd like to build the set of all possible car versions.
The usual ordered pair (x,y) = {{x}, {x,y}}.
Your question is not about sets; it is about multisets or bags.
A simulation of a bag for your problem is
T×{0,1} $\cup$ S×{0},
where T is the set of two version elements
and S is the set of one version elements.