ITPEC FE Morning October 2020 Question 25

{A, P} — together A and P determine every attribute of R, and neither does on its own.

A and P never appear on the right-hand side of a dependency, so nothing can derive them: every key must contain both. Compute the closure of {A, P}:

• •A → B and A → C add B and C
• •With C and P now known, CP → Q and CP → T add Q and T
• •The closure is {A, B, C, P, Q, T}, the whole relation

Dropping either attribute fails: A alone gives only {A, B, C}, and P alone gives nothing new. So {A, P} is a minimal key.

Answer (c)

Why not others:

• •(a) A — its closure {A, B, C} lacks P, so CP → Q and CP → T never fire and P, Q, T are missed
• •(b) A, B — B adds nothing A does not already give, and P is still missing, so the closure is again {A, B, C}
• •(d) C, P — its closure {C, P, Q, T} lacks A and B, because nothing determines A

Key rule: An attribute that never appears on the right-hand side of any functional dependency belongs to every key; start from those and check whether their closure already covers the relation.

Databases · AI-written, may contain errors