Question

A – ( B ∩ A )

I was wondering what this set of mathematics can translate to when looking at,comparing it with SQL (operators).

Answer 1

If A and B are tables of the same “type” (same number of columns and compatible datatypes on corresponding columns), this can be translated to SQL like this:

A EXCEPT (A INTERSECT B) 

which is of course equivalent (both in set and relational algebra) to:

A EXCEPT B 

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If A and B are tables of different “type”, then set operations do not make sense between them. (Joins are something different, they should not be confused with Unions, Differences or Intersections, no matter how popular a link that “explains” them is.)

The fact that Unions, Differences and Intersections can also be expressed in several ways (using (LEFT) JOIN, (NOT) IN, (NOT) EXISTS combinations, besides the explicit UNION, EXCEPT and INTERSECT operators) does not change that.

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The syntax is not exactly as above. One can use either (works in Postgres and SQL-Server. It also works in Oracle if one replaces EXCEPT with MINUS):

  SELECT *
  FROM a

EXCEPT 

  ( SELECT * 
    FROM a
  INTERSECT
    SELECT *
    FROM b
  ) ;

or this (works in Postgres 8.4 and above: SQL-Fiddle test)

SELECT *
FROM
  ( TABLE a
  EXCEPT 
    ( TABLE a INTERSECT TABLE b ) 
  ) t  ;

and even this (Look ma, no SELECT!):

 TABLE a
EXCEPT 
  ( TABLE a  INTERSECT  TABLE b ) ;