Set Unions with repeating elements the union of {1,2,3,4} and {2,3,4,5} is {1,2,3,4,5}
but the union of {2,2,2,3} and {2,3,3,5} is {2,3,5}
Is there another Union concept which takes account of the number of occurences of repeating elements. i.e. 
the SOME_TERM of {2,2,2,3} and {2,3,3,5} is {2,2,2,3,3,5}. basically the intersection of the set would only account for one of each repeating term in both sets.
 A: There isn't such a term in common use, to the best of my knowledge, but the concept is well known. (For example, the least common multiple of $2\times 2\times 2\times 3$ and $2\times 3\times 3\times 5$ is exactly $2\times 2\times 2\times 3\times 3\times 5$!)
The reason such a term isn't in common use is that sets don't take account of multiplicity anyway: that is, {2, 2, 3} = {2, 3} = {2, 2, 2, 1+1, 3, 3, 3, 2, 3}. Sets only care about what elements they contain.
A: You could define a new datat-type for your set $S$, which contains 2-tuples such that the first element is the "object" you wish contain (in this case numbers, perhaps even integers, though you could relax that constraint to contain anything you like, even other sets!) and the second element is the count of how many times that element is in the set. So your first set would look like:
$$
\{(1,1),(2,1),(3,1),(4,1)  \}
$$ 
and so on with the others. 
Note that this formulation allows for the counter-intuitive existence of "0-count" elements in a set, such as $(1,0)$, though we shall soon see that they won't pose a problem.
Then rather than considering a normal union, we can instead define a new operation, $\cup_{new}$ which adds multiplicities, so that we would have something along the lines of:
$$
\{(2,3), (3,1)  \} \cup_{new} \{(2,1),(3,2),(5,1)  \} = \{(2,4), (3,3), (5,1)  \}.
$$
A more general formulation of this concept is possible as well. 
Considering the "0-count" elements, by the additive identity, we have that objects like $\{(x,0)\} \cup_{new} \{(x,n)\} = \{(x,n)\}$. where $n$ is a natural number or 0, and $x$ is one of your set objects (again, in this case integers).
Taking a similar, approach, we could define a new operation $\cap_{new}$, which performs the sort of operation you desire.
