Yes indeed, Intersection, Complement and Union force it to be mathematical sets, while Subsets is not so stringent.

Union removes doubles, but DeleteDuplicates does it a lot faster!

I think you could easily do DeleteDuplicates on it to eliminate 'doubles'. I.e. now the function is more useful, as you would not be able to do the opposite... Why Union/Complement/Intersection delete duplicates and sort the output? Just a design choice I think... tricky stuff indeed...

OK, thanks. That's a simple resolution.

Jeff

I think it is consistent if you read the guide page (guide/OperationsOnSets):

If you want to keep duplicates I suggest you use something like:

DeleteCases[eall,Alternatives@@Join[e1,e2,e3,...]]

Sorry, but I don't understand that question!

From a strictly mathematical point of view, each object is a set (provided, of course, you stay away from objects that are "too big" to be considered sets and would fall within the category of "proper classes"). In particular, a "list" is modeled mathematically as an "ordered set", which more precisely can be defined as a function from an ordered index set to the set of elements that form the "entries" of the list.

In Mathematica, a list is such an ordered set, so that, for example, {a, a, a, b, c} is not the same thing as the list {a, b, c} — the latter being the result of Union[{a, a, a, b, c}].

Unfortunately, Mathematica does not directly provide a built-in kind of object that models sets. Perhaps the confusion is the use in Mathematica of the curly brace notation {...} notation for a list, whereas that same notation is conventionally used in mathematics to denote a set (which need have no order associated with its elements).

Just to make things worse, many mathematicians, and many mathematics books (including, alas, many if not most calculus books) use the curly brace notation for ordered sets, particularly when they are denoting sequences.

Elements of a set are not the same thing as entries in a list. The docs indicate that Complement treats the first argument, a list, as if it were a set and then deletes from that set those elements that belong to the second argument (or subsequent arguments). And that's what the result shows.

Of course it doesn't matter that the example you give has entries that are single-character strings. The same thing would result if, assuming no values had been assigned to a, b, c, or d, you evaluated instead Complement[{a, a, a, b, c}, {c, d}] to obtain result {a, b}.

If you really want to preserve the multiplicity of entries in the first argument, you may do so like this:

      DeleteCases[{a, a, a, b, c}, Alternatives[c, d]]
(* {a, a, a, b} *)

But it is deleting elements. Here is the actual code and result:

Complement[{"a","a","a","b","c"},{"c","d"}] {a,b}

Jeff