# The Set/Bag Irrelevance

From time to time it is discussed whether SQL’s support for duplicate tuples in queries is useful and Codd et al’s relational model is lacking because it forbids them, or whether they make SQL a worse language and the relation model is better off for eliminating them. The former could be called “allowing bags”. Bags (or multisets) are like sets except they permit repeated elements.

Date discusses the issue in the 22 pages of the “Date on Database” chapter “Double Trouble, Double Trouble”. His objections to duplicate tuples seem to be (paraphrasing)

1. “The existence of duplicate tuples implies the ability to distinguish identical things”. This objection I do not understand. He relates it vaguely to a philosophical concept of “The Principle of Indistinguishability of Indiscernables” though the reasoning is unclear. Date claims (perhaps correctly) that there is no way to distinguish the “two” 6s that appear in the bag {3,6,6,8,8,8,11} but again it is unclear why any particular conclusion should follow from this.

2. “Relations are sufficient thus bags should be excluded for the sake of parsimony”. This argument I am susceptible to, but it crucially hinges on relations actually being sufficient. More on this later; it’s the crux of the issue!

## Sets and bags are equivalent

Despite all this handwringing it turns out that in a decent relational query language (Date and Darwen might say “in a D”) having set semantics is exactly the same as having bag semantics. That is, bags can be implemented in terms of sets and vice versa. This shouldn’t be surprising really but I don’t think the consquences of this observation have really been thought through.

Here’s how you show the (trivial) correspondence. Our decent relational query language will have type abstraction so whereas Date wants us to only use Set (a, b, c, ...) to represent a relation on a, b, c, … we are welcome to define

type Bag t = Set (t, N+)

where N+ is some previously defined domain type of strictly positive integers. With a bit of fiddling we can define conversions from Set t to Bag t and vice versa (an adjunction or at least adjunction-like) as well as project, extend, restrict, join and indeed all our relational operators for the Bag type. (Union is an interesting case. There are at least two sensible concepts of union (sum and max) and it would probably be practical to provide both).

If on the other hand we were living in Date’s nightmare bag-world we would define

type Set t = Bag t KEY t

That is, we define a Set to be be a Bag where the whole tuple is a key. This implies that there can exist at most one tuple of any given value in the multiset, i.e. it is in fact a set. As before we can implement all the functionality for sets in terms of existing functionality for bags.

## The argument is irrelevant

Imagine that the major proponent of a call by value lambda calculus claims that the only tupling operation must be binary. Detractors claim “no we must have tuples of arbitrary size”. Once you have one you have the other. There’s no point trying to forbid one option.

Thus we see that the argument about sets and bags is irrelevant. If we have a decent query language that allows some degree of type abstraction we automatically have both if we have either. It doesn’t matter whether Date thinks that Occam’s razor implies we should work with sets only. We’ll get bags too whether he likes it or not.