Relational Algebra equivalent of SQL "NOT IN"

Question

Is there a relational algebra equivalent of the SQL expression NOT IN?

For example if I have the relation:

A1  |  A2
----------
x   |  y
a   |  b
y   |  x

I want to remove all tuples in the relation for which A1 is in A2. In SQL I might query:

SELECT
    *
FROM
    R
WHERE
    R.A1 NOT IN
        (
        SELECT
            A2
        FROM
            R
        )
/

What is really stumping me is how to subquery inside the relational algebra selection operator, is this possible?:

σ_{some subquery here}R

Answer 1

In relational algebra, you can do this using a carthesian product. Something like:

R - ρ_a1,a2(π_a11,a21(σ_{A11 = A22}(ρ_a11,a21(R) x ρ_{a12, a22}(R))))

• rename the columns of R, f.e. from a1 to a11 (left hand) and a12 (right hand)
• take the cross product of the R's with renamed columns
• select rows where a11 equals a22
• project out a12 and a22 and keep a11 and a21
• rename to a1 and a2

That gives you the rows that were matched. Subtract this from R to find the rows that where not matched.