How to find all tuples in a table if and only if the tuple appears once?

Question

I have a table:

x | y | z
------------
1 | 1 | *
1 | 1 | *
1 | 3 | *
2 | 2 | *
2 | 3 | *
3 | 4 | *
3 | 4 | * 
3 | 3 | *

What is the relational algebra representation of only returning all unique (x, y) tuples?

For example, I would like the following (x,y) tuples returned in the above table: (1,3), (2,2) (2,3), and (3,3).

Thanks

Answer 1

1. Rename R to S

   S := ρ_S/R(R)

2. Join R and S on x,y

   D := R ⋈_{S.x = R.x ∧ S.y = R.y} S

   This squares the number of tuples with a particular value for (x,y). Particularly, if a value for (x,y) appears only once in R, it appears only once in D.

3. Join R and S on x,y,z

   E := R ⋈_{S.x = R.x ∧ S.y = R.y ∧ S.z = R.z} S

   This basically adds some columns to R. It does not add or remove tuples.

4. Subtract E from D and project to the attributes of R

   F := π_x,y,z(D\E)

   This removes the tuples from D, that where created by joining a tuple from R to the corresponding tuple in S. The remaining tuples are the ones that where created by joining a tuple in R to a different tuple in S. Particularly, if a value for (x,y) appears only once in R, no tuple in F exists with that value.

5. Remove the tuples in F from R

   R\F