GATE CS 2021 Set 2 Q40 — Normal Forms

To determine which functional dependencies can be inferred, we compute the attribute closures of the left-hand sides of the options using the given functional dependencies:

1. $P \rightarrow QR$
2. $RS \rightarrow T$

Let's evaluate each option:

• Option A ($PS \rightarrow T$): Computing the closure of $PS$: $$(PS)^+ = \{P, S\} \xrightarrow{P \rightarrow QR} \{P, Q, R, S\} \xrightarrow{RS \rightarrow T} \{P, Q, R, S, T\}$$ Since $T \in (PS)^+$, $PS \rightarrow T$ holds. Hence, Option A is correct.
• Option B ($R \rightarrow T$): Computing the closure of $R$: $$(R)^+ = \{R\}$$ Since $T \notin (R)^+$, $R \rightarrow T$ does not hold. Hence, Option B is incorrect.
• Option C ($P \rightarrow R$): Computing the closure of $P$: $$(P)^+ = \{P\} \xrightarrow{P \rightarrow QR} \{P, Q, R\}$$ Since $R \in (P)^+$, $P \rightarrow R$ holds. Hence, Option C is correct.
• Option D ($PS \rightarrow Q$): As computed above, $(PS)^+ = \{P, Q, R, S, T\}$. Since $Q \in (PS)^+$, $PS \rightarrow Q$ holds. Hence, Option D is correct.