GATE CS 2026 Set 1 — Question 30
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Databases → Normalization → Armstrong's Axioms
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Question
Let and be the attributes of a relation in a relational schema. Let indicate functional dependency in the context of a relational database, where . Which of the following options is/are always true?
Correct answer
(C) If (\P\ \R\ and \Q\ \S\), then \P, Q\ \R, S\; (D) If \P\ \R\, then \P, Q\ \R\
Solution
To determine which options are always true, we apply Armstrong's Axioms and derived rules for functional dependencies (FDs):
1.Option (C) is true by the Composition Rule: If
X → Y and W → Z, then XW → YZ. - Given
P → RandQ → S, applying the composition rule directly gives .
X → Y, then XZ → YZ for any . - Given
P → R, augmenting with givesPQ → RQ. - By the Decomposition Rule (or reflexivity
RQ → Rand transitivity), ifPQ → RQ, thenPQ → Rmust hold.
P → R holds, and by augmentation PQ → R also holds. However, Q → R does not necessarily hold as might not determine .4.Option (B) is false. This describes a partial dependency. In a normalized relation, an attribute might depend on a composite key without depending on either or individually (e.g., ).
Therefore, options (C) and (D) are always true.Continue learning with Success Tracker
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