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GATE 2026 CS (CS2) – Question 42

Databases · Integrity Constraints and Normal Forms · 2 marks · Multiple choice

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 $F$. Armstrong's axioms are Reflexivity, Augmentation, and Transitivity. The derived Union rule is: if $X \to Y$ and $X \to Z$, then $X \to YZ$. Which one of the following combinations of Armstrong's axioms is both necessary and sufficient to prove the Union rule?

  1. Reflexivity, Augmentation, and Transitivity
  2. Reflexivity and Augmentation
  3. Transitivity
  4. Augmentation and Transitivity

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Correct answer: (D) Augmentation and Transitivity

Explanation

To derive Union from $X\to Y$ and $X\to Z$, first apply Augmentation to $X\to Y$ using $X$ to get $X\to XY$. Next apply Augmentation to $X\to Z$ using $Y$ to get $XY\to YZ$. Then apply Transitivity to conclude $X\to YZ$. Reflexivity is not needed. Therefore, Augmentation and Transitivity are sufficient, and they are the required combination. Hence option (D) is correct.