GATE CS 2025 Set 1 | Question 47

MSQ · 2 marks

Consider a relational schema \(team(name, city, owner)\), with functional dependencies \(\{name \to city,\ name \to owner\}\).

The relation \(team\) is decomposed into two relations, \(t1(name, city)\) and \(t2(name, owner)\). Which of the following statement(s) is/are TRUE?

  1. Option A: The relation \(team\) is NOT in BCNF.
  2. Option B: The relations \(t1\) and \(t2\) are in BCNF. (correct)
  3. Option C: The decomposition constitutes a lossless join. (correct)
  4. Option D: The relation \(team\) is NOT in 3NF.

Answer

(B), (C)

Explanation

  • Key: \(name^+ = \{name, city, owner\}\), so \(name\) is a candidate key of \(team\). Both FDs have this key on the left, so \(team\) is already in BCNF, and therefore in 3NF too. So (A) and (D) are false.
  • (B) True: in \(t1\) the only FD is \(name \to city\) and in \(t2\) it is \(name \to owner\); in each, \(name\) is a key. (Any relation with just two attributes is in BCNF anyway.)
  • (C) True: the common attribute of \(t1\) and \(t2\) is \(name\), which is a key of \(t1\) (and of \(t2\)), so joining them on \(name\) gives back exactly the original relation.

Answer: B, C