---
title: "Lossless and dependency preserving check 3"
subject: "Database Management System"
chapter: 4
key: "semester_6/database_management_system:decomp-abc-ab-ac"
source: https://www.bctengineer.com/q/semester_6/database_management_system/decomp-abc-ab-ac
site: BCT Engineering
---

# Lossless and dependency preserving check 3

Database Management System · Chapter 4: Database Constraints & Normalization

- Asked in: year not given
- Video: [Illustration on Lossless Join Decomposition and Dependency Preserving Decomposition. | DBMS](https://www.youtube.com/watch?v=iAarosX8ees) — Edredo for Learners
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/database_management_system/decomp-abc-ab-ac
- Whole chapter: https://www.bctengineer.com/md/semester_6/database_management_system/chapter-4

## Question

R (A, B, C), R1 (A, B), R2 (A, C). F = {A → B, B → C}. Check whether the decomposition is lossless and dependency preserving.

## Short answer

The decomposition is lossless because common attribute A is a key of R. It is not dependency preserving: F1 ∪ F2 yields only A→B and A→C, so B→C is lost.

## Answer

R = (A, B, C), R1 = (A, B), R2 = (A, C), F = {A → B, B → C}.

#### Lossless join test
R1 ∩ R2 = {A}. A⁺ = {A, B, C} = R (A → B, then B → C). So A is a key of R, and hence A → R1 and A → R2.

Chase table:

| | A | B | C |
|---|---|---|---|
| R1 | a | a | b |
| R2 | a | b | a |

A → B: both rows have A = a, so B in R2 becomes a. Then B → C: both rows have B = a, so C in R1 becomes a. Both rows are now all "a".

**The decomposition is lossless.**

#### Dependency preservation test
Projections:
- F1 on (A, B): A → B
- F2 on (A, C): A → C (obtained from A → B, B → C)

Together F1 ∪ F2 = {A → B, A → C}.

- A → B: in F1. Preserved.
- B → C: B and C are in different relations. Test: Z = {B}; R1: B⁺ ∩ {A, B} = {B}; R2: Z ∩ R2 = ∅. C is never reached. Under {A → B, A → C}, B⁺ = {B}. **Lost.**

**The decomposition is NOT dependency preserving.**

**Answer: lossless, but not dependency preserving.** Example: R1 can hold (a1, b1), (a2, b1) and R2 can hold (a1, c1), (a2, c2). Each table is consistent with A → B and A → C, but the join has b1 with both c1 and c2, violating B → C, and no single table can detect it.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For R1 = (A,B), R2 = (A,C) with F = {A->B, B->C}, why is the decomposition lossless?
   A. A+ = {A,B,C} = R, so the common attribute A is a key
   B. B+ = {B,C}
   C. R1 and R2 are the same size
   D. Because A is in both relations
   Answer: A. R1 and R2 share A, and A determines all of R, so the lossless condition holds. (Key point: Lossless test)

2. Which FD is lost in this decomposition?
   A. A -> B
   B. A -> C
   C. B -> C
   D. A -> BC
   Answer: C. B and C are in different relations, and with {A->B, A->C} the closure of B is only {B}. (Key point: Lost dependency)

3. What is the practical problem when B -> C is not preserved?
   A. The join cannot be computed
   B. Each table stays consistent but the join can violate B -> C and no single table detects it
   C. Data is lost during the join
   D. A becomes a non-key
   Answer: B. For example b1 can pair with c1 and c2 in the join, violating B -> C, and checking it needs a join. (Key point: Meaning of lost FD)

## Close variants set in other exams

- [Lossless and dependency preserving check 2](https://www.bctengineer.com/question.md?key=semester_6%2Fdatabase_management_system%3Adecomp-abcdeg-four-relations) — Database Management System, chapter 4
- [Lossless and dependency preserving check 4](https://www.bctengineer.com/question.md?key=semester_6%2Fdatabase_management_system%3Adecomp-abcde-abc-ade) — Database Management System, chapter 4
- [Lossless and dependency preserving check 5](https://www.bctengineer.com/question.md?key=semester_6%2Fdatabase_management_system%3Adecomp-abcd-abc-cd) — Database Management System, chapter 4

## Related in Database Management System

- [Lossless and dependency preserving check 1](https://www.bctengineer.com/question.md?key=semester_6%2Fdatabase_management_system%3Adecomp-abcde-r1-r2-cde) — Database Management System, chapter 4

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