# Relational Division in SQL The Easy Way

I recently studied SQL as part of an introductory course on databases. SQL itself is not particularly difficult to grasp, yet compared to relational algebra, the division operation is much more complex. In relational algebra, there is a division operator, which has no direct equivalent in SQL. This means that you’ll have to find a workaround. There are a number of ways to express division in SQL, and with the exception of one, they are all quite complex.

Division identifies attribute values from a relation that are paired with all of the values from another relation. Popular textbook examples are the identification of suppliers who deliver all parts of a particular color. An intuitive solution would be to count the number of distinct red parts, and then look at every distributor to find out which of those deliver all those parts. To express this in SQL, you have to use the set theoretic operators “having” and “group by”, and then you simply count the tuples meeting certain criteria.

Let’s say you have table T1 in front of you and want to find out which A’s have both b2 and b3. You can assume that b2 and b3 are the red parts. If you’ve only been exposed to standard textbook treatments of division in SQL, you may be surprised that the problem can be solved as simply as this:

```SELECT A
FROM T1
WHERE B IN (
SELECT B
FROM T2
)
GROUP BY A
HAVING COUNT(*) = (
SELECT COUNT (*)
FROM T2
);
```

Of course you can add a Where clause to the last expression.

The previous example is quite easy to grasp. The same can’t be said about how SQL division is commonly taught. Standard database theory textbooks expose you to a statement that is doubly nested and peppered with two negations. No matter how smart you are, it takes longer to parse than the previous example. I think you can safely call the following a monstrosity:

```SELECT DISTINCT x.A
FROM T1 AS x
WHERE NOT EXISTS (
SELECT *
FROM	T2 AS y
WHERE NOT EXISTS (
SELECT *
FROM T1 AS z
WHERE (z.A=x.A) AND (z.B=y.B)
)
);
```

Which one would you prefer?

The examples above are taken from the paper “A Simpler (and Better) SQL Approach to Relational Division” by Matos and Grasser, published in Journal of Information Systems Education, Vol. 13(2). I was quite happy to have come across that paper. Unfortunately, theirs is not a very well-known approach to SQL division. This is unfortunate, since it’s not only easier to grasp, but, as Matos and Grasser write, it also exhibits better computational performance.