ITPEC FE Morning April 2020 Question 28

Relational Division — find Y values in R that are associated with ALL X values in S.

Given expressions:
- T₁ ← πY(R) → project Y from R → {A, B}
- T ← T₁ − πY((S × T₁) − R)

Step-by-step:

• •S × T₁ = all combinations of S.X and T₁.Y:
• •{(1,A), (1,B), (2,A), (2,B)}
• •(S × T₁) − R = combinations NOT in R:
• •R = {(1,A), (2,A), (2,B)}
• •Result: {(1,B)}
• •πY(...) = {B} — Y values that are MISSING some X
• •T = T₁ − {B} = {A, B} − {B} = {A}

Answer: (b)

Why not others:
- (a) NULL — A satisfies the division; result is not empty
- (c) B — B is only paired with X=2, not with X=1
- (d) {A, B} — B fails the completeness check

Key rule: T₁ − πY((S × T₁) − R) is the standard relational division formula. It answers: "Which Y values in R are associated with every X value in S?"

AI-generated — may contain errors