ITPEC FE Morning April 2023 Question 26

OS — O and S together determine every attribute of R, and neither does alone.

O and S never appear on the right side of a dependency, so nothing else can derive them and every key must contain both. Their closure:

• •Start with O and S
• •O → T adds T
• •S → M adds M
• •OS → P adds P
• •M → N adds N, so all six attributes are reached

Dropping O loses T and P; dropping S loses M, N and P. So OS is minimal, a candidate key.

Answer (d)

Why not others:

• •(a) MO — reaches only M, N, O, T; without S, P cannot be derived.
• •(b) MS — reaches only M, N, S; without O neither T nor P follows.
• •(c) OP — reaches only O, P, T; without S neither M nor N follows.

Key rule: An attribute that never appears on the right side of any functional dependency belongs to every candidate key; start from those attributes and compute their closure.

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