How many minimum relations are required for the following Relation R(A, B, C,…
How many minimum relations are required for the following Relation R(A, B, C, D, E) with FD {A → BC, CD → E, B → D, E → A} to convert into BCNF without violation of lossless and dependency preserving decomposition _________.
Answer: D. 4 — Original FDs: A→BC, B→D, CD→E, E→A. BCNF decomposition (dependency‑preserving & lossless): 1. R1(ABC) for A→BC 2. R2(BD) for B→D 3. R3(CDE) for CD→E 4. R4(EA)…
- A.
1
- B.
2
- C.
3
- D.
4
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Correct answer: D
Original FDs: A→BC, B→D, CD→E, E→A.
BCNF decomposition (dependency‑preserving & lossless):
1. R1(ABC) for A→BC
2. R2(BD) for B→D
3. R3(CDE) for CD→E
4. R4(EA) for E→A
Why not merge R1 and R4 → R(ABCE)?
Projected FDs in ABCE: A→BC, E→A (⇒ E→ABC).
Check BCNF: every FD’s LHS must be a superkey.
For A→BC, A is NOT a superkey of ABCE (A⁺ = {A,B,C} ≠ ABCE).
Hence ABCE violates BCNF.
BD and CDE are BCNF (B and CD are keys in their relations).
The 4-relation set preserves all FDs and is lossless; the 3-relation set (BD, ABCE, CDE) is not BCNF.
Final: Minimum relations for BCNF, lossless, dependency‑preserving = 4.