Inference from functional dependencies — evaluate the claim: “Given the dependency R → (S, T), it also follows that R → S.” Decide whether this inference is correct or incorrect.
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ACorrect — by the decomposition rule of functional dependencies
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BIncorrect — you need R → T before inferring R → S
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CCorrect only if R is a candidate key
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DNot enough information — sample data required
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EIncorrect — composition rather than decomposition applies
Answer
Correct Answer: Correct — by the decomposition rule of functional dependencies
Explanation
Introduction / Context: Reasoning with functional dependencies relies on inference rules (Armstrong’s axioms and derived rules). One of the most used rules is decomposition (also called projectivity), which allows splitting a dependency with multiple attributes on the right-hand side.
Given Data / Assumptions:
- We are given R → (S, T), where R, S, and T are attribute sets.
- We assume standard FD semantics and closure rules.
- No special assumptions about keys are needed.
Concept / Approach: The decomposition rule states: If X → YZ then X → Y and X → Z. This follows from reflexivity and augmentation within Armstrong’s axioms. Intuitively, if R determines both S and T together, then R determines S individually (and T individually) as well.
Step-by-Step Solution:
Start with R → ST (Y = S, Z = T).Apply decomposition: derive R → S and R → T.Therefore, the claim R → S is valid.No key assumption is necessary for this inference.Verification / Alternative check: Compute attribute closure R+ given R → ST: S ∈ R+, T ∈ R+. Hence R → S is implied.
Why Other Options Are Wrong:
- Requiring R → T first is unnecessary; both are simultaneous consequences.
- Key status is irrelevant to the inference.
- Sample data is not required; inference is rule-based.
Common Pitfalls: Thinking combined determination is stronger than individual determination; forgetting standard FD inference rules.
Final Answer: Correct — by the decomposition rule of functional dependencies