How to query a nested document in mongo?

How can we mathematically prove that a nested SQL query cannot be unnested?

• For example, the following nested SQL query cannot be unnested: SELECT AVG(salary) FROM ( SELECT DISTINCT department, salary FROM emp )

• Answer:

  Assume department1 has 10 employees with salary "$1".  department2 has 25 employees with salary "$2". Average value over 10 × $1 + 20 × $2 = $50 / 30 employees = $1.66 But if we only count one instance of each distinct department & salary, then: Average value over 1 × $1 + 1 × $2 = $3 / 2 employees = $1.5 You can try SELECT AVG(DISTINCT salary), but this is yet another way of calculating the average:  if there's a department3 which also has salary "$2" it wouldn't count toward the average because it would be subsumed by the same salary value in department2.  Each distinct salary value is counted only once, regardless of how many departments it occurs in. Re your comment: I was using a proof by http://en.wikipedia.org/wiki/Counterexample.  This is a common way to invalidate a logical statement. I'm not sure we can come up with an easy hint that a nested query needs to stay nested.  You just have to follow the logic and see if would be guaranteed to give the same results if it were not nested.