Intersecting a dynamic number of tables - sql

Question

I have two relations, a and b, with attributes given by
CREATE TABLE a (id int, b_id int)
CREATE TABLE b (id int)
for which I can assume that all pairs of values in a and all values in b are unique, and which will be based in an SQL Server 2016 database.
A given element of b defines a subset of a.id given by those elements for which the corresponding a.b_id is the given value, and my goal is to produce the intersection of all those subsets.
Say, for instance, that a contains the six values,
INSERT INTO a VALUES (1, 1), (1, 2), (1, 3), (2, 2), (3, 2), (3, 3)
Then the expected results would include the following:
b: (1), (2), (3). Expected result: (1)
b: (1), (2). Expected result: (1)
b: (2). Expected result: (1), (2), (3)
b: (2), (3). Expected result: (1), (3)
b: [Empty]. Expected result: (1), (2), (3)
Using uniqueness, for the case of non-empty b, this can be achieved through
SELECT a.id FROM a
JOIN b on a.b_id = b.id
GROUP BY a.id
HAVING COUNT(a.id) = (SELECT COUNT(*) FROM b)
but this feels clunky, given that SQL has an INTERSECT operator readily available, and were I to write the same query in, say, LINQ, I would simply aggregate intersections. It also fails to produce the desired result in the case of empty b without treating that as a special case.
So, the question becomes: Is there a more idiomatic way of performing the above query, which also works properly for trivial b?

Answer 1

What you are trying to do is called a relational division [1, 2].
CREATE TABLE a (id INT, b_id INT);
CREATE TABLE b (id INT);
INSERT INTO a VALUES
(1, 1), (1, 2), (1, 3),
(2, 2), (3, 2), (3, 3);
DECLARE #i INT = 0;
WHILE #i < 5 BEGIN
TRUNCATE TABLE b;
IF #i = 0 INSERT b VALUES (1), (2), (3);
IF #i = 1 INSERT b VALUES (1), (2);
IF #i = 2 INSERT b VALUES (2);
IF #i = 3 INSERT b VALUES (2), (3);
SELECT DISTINCT x.id
FROM a AS x
WHERE NOT EXISTS (
SELECT *
FROM b AS y
WHERE NOT EXISTS (
SELECT *
FROM a AS z
WHERE z.id = x.id AND z.b_id=y.id
)
)
SET #i = #i + 1;
END;
Test it online.

Answer 2

You could extend your approach to handle empty set case by using:
SELECT a.id
FROM a
LEFT JOIN b on a.b_id = b.id
GROUP BY a.id
HAVING COUNT(b.id) = (SELECT COUNT(*) FROM b);
DBFiddle Demo | DBFiddle Demo - all test cases
Extra: Transient Data (used in second demo)
EDIT:
Another approach to handle empty set and leave INNER JOIN:
SELECT a.id FROM a
JOIN b on a.b_id = b.id
GROUP BY a.id
HAVING COUNT(a.id) = (SELECT COUNT(*) FROM b)
UNION
SELECT a.id
FROM a
WHERE NOT EXISTS (SELECT 1 FROM b);
DBFiddle Demo 3