On Sat Feb 07 2015 at 11:12:02 PM Neil Girdhar <
mistersheik@gmail.com> wrote:
Because yielding an ordering doesn't have any basis in the definition of union, and it's not any easier to write than a + b + c + d. "copy and extend" is just concatenation - is there any reason *not* to use the concatenation operator for it, rather than the union?
In a+b, I'd expect that all of the elements of b will appear in the result in the same order they appear in b. That's how it works for sequences. But it can't work that way for ordered sets if there are any elements of b also in a.
More importantly, that's the entire point of ordered sets. If you wanted to preserve (or didn't care about) duplicates, you'd just use a list.
What you actually want here is an "unsorted union"--all the elements of a in order, then all the elements of b except those also in a in order, and so on. (That's "unsorted" as opposed to a sorted union, where the sequences are interleaved in such a way as to preserve a shared ordering rule.) It seems a lot more natural to call that "union" than concatenate. But it might be better to explicitly call it "unsorted union", or even to just not provide it. (IIRC, in Mathematica, you do it by concatenating the sets into a list, then calling DeleteDuplicates to turn the result back into an ordered set, or by calling Tally on the concatenated list and selecting the ones whose tally is 1, or various other equivalents.)
The fact that unordered union isn't commutative doesn't mean it's not a reasonable extension of union. The best-known extension of union is probably the disjoint union, which also isn't commutative: {1, 2} U* {0, 1} = {(1,0), (2,0), (0,1), (1,2)} but {0, 1} U* {1, 2} = {(0, 0), (1, 0), (1, 1), (2, 1)}.