GATE CS 2026 Set 2 — Question 42
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Databases → Normalization → Armstrong's Axioms
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Question
In the context of schema normalization in relational DBMS, consider a set F of functional dependencies. The set of all functional dependencies implied by F is called the closure of F. To compute the closure of F, Armstrong's Axioms can be applied. Consider and as sets of attributes over a relational schema. The three rules of Armstrong's Axioms are described as follows.Reflexivity: If , then
Augmentation: If
Transitivity: If
Union: If
X → YAugmentation: If
X → Y, then XZ → YZ for any Transitivity: If
X → Y and Y → Z, then X → ZThe additional rule of Union is defined as follows.Union: If
X → Y and X → Z, then X → YZIt can be proved that the additional rule of Union is also implied by the three rules of Armstrong's Axioms. Listed below are four combinations of these three rules. Which one of these combinations is both necessary and sufficient for the proof?Correct answer
(D) Augmentation and Transitivity
Solution
The proof for the Union rule () using Armstrong's Axioms is as follows:
1.
X → Y (Given)2.
X → Z (Given)3.Apply Augmentation to (1) with :
(Note: Strictly, Augmentation gives XX → XY, which simplifies to X → XY.XZ → YZ. If we augment X → Y with , we get XX → YX, i.e., X → XY.)4.Apply Augmentation to (2) with :
XY → ZY, which is XY → YZ.5.Apply Transitivity to (3) and (4): Since
Thus, Augmentation and Transitivity are sufficient and necessary to derive the Union rule from the base axioms.X → XY and XY → YZ, then X → YZ.Continue learning with Success Tracker
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