Question

Let 𝑃, 𝑄, 𝑅 and 𝑆 be the attributes of a relation in a relational schema. Let 𝑋 ⟶𝑌
indicate functional dependency in the context of a relational database, where
𝑋, 𝑌 ⊆{𝑃, 𝑄, 𝑅, 𝑆}.
Which of the following options is/are always true?

• A. If ( {𝑃, 𝑄} ⟶{𝑅} and {𝑃} ⟶{𝑅} ), then {𝑄} ⟶{𝑅}
• B. If {𝑃, 𝑄} ⟶{𝑅}, then ( {𝑃} ⟶{𝑅} or {𝑄} ⟶{𝑅} )
• C. If ( {𝑃} ⟶{𝑅} and {𝑄} ⟶{𝑆} ), then {𝑃, 𝑄} ⟶{𝑅, 𝑆}
• D. If {𝑃} ⟶{𝑅}, then {𝑃, 𝑄} ⟶{𝑅}

Hint:

Use Armstrong's axioms: augmentation always lets you add extra attributes to the left side of a dependency (option D), and combining two independent dependencies via augmentation plus transitivity gives their union (option C); the reverse-direction splits in options A and B are not generally valid.

Answer 1

The Correct Option is C, D

Use Armstrong's axioms, reflexivity, augmentation, and transitivity, to test each claim about functional dependencies over attributes P, Q, R, S.

The first claim says \(PQ \to R\) and \(P \to R\) together force \(Q \to R\). But \(P \to R\) already explains why \(PQ \to R\) holds by augmentation alone, so this tells us nothing about whether Q by itself determines R. A relation where R depends on P but is unrelated to Q satisfies both premises while \(Q \to R\) fails, so this claim is not always true.

The second claim says \(PQ \to R\) forces either \(P \to R\) or \(Q \to R\) individually. Functional dependencies can genuinely require the combination of attributes on the left side, for instance R could be computed jointly from both P and Q with neither alone pinning down R. So this claim is not always true.

The third claim says \(P \to R\) combined with \(Q \to S\) forces \(PQ \to RS\). This is derivable: augment \(P \to R\) with Q to get \(PQ \to RQ\), augment \(Q \to S\) with R to get \(RQ \to RS\), then chain by transitivity to get \(PQ \to RS\). Since this is a formal consequence of the axioms, it always holds, so this claim is true.

The fourth claim says \(P \to R\) forces \(PQ \to R\). This is exactly the augmentation axiom applied with an extra attribute Q added to both sides, which is always valid, so this claim is true.

Hence the two dependencies that are always guaranteed are the third and fourth claims, options C and D.