# Lossless Decomposition in DBMS

Lossless join decomposition is a decomposition of a relation R into relations R1, R2 such that if we perform a natural join of relation R1 and R2, it will return the original relation R. This is effective in removing redundancy from databases while preserving the original data…

In other words by lossless decomposition, it becomes feasible to reconstruct the relation R from decomposed tables R1 and R2 by using Joins.

Only 1NF,2NF,3NF and BCNF are valid for lossless join decomposition.

In Lossless Decomposition, we select the common attribute and the criteria for selecting a common attribute is that the common attribute must be a candidate key or super key in either relation R1, R2, or both.

Decomposition of a relation R into R1 and R2 is a lossless-join decomposition if at least one of the following functional dependencies are in F+ (Closure of functional dependencies)

R1 ∩ R2 → R1 OR R1 ∩ R2 → R2

**Question 1:**

Let R (A, B, C, D) be a relational schema with the following functional dependencies:

A → B, B → C, C → D and D → B. The decomposition of R into (A, B), (B, C), (B, D)

**(A)** gives a lossless join, and is dependency preserving **(B)** gives a lossless join, but is not dependency preserving **(C)** does not give a lossless join, but is dependency preserving **(D)** does not give a lossless join and is not dependency preserving

Refer to this for a solution.

**Question 2 **

R(A,B,C,D) is a relation. Which of the following does not have a lossless join, dependency preserving BCNF decomposition?

(A) A->B, B->CD

(B) A->B, B->C, C->D

(C) AB->C, C->AD

(D) A ->BCD

Refer to this for a solution.

Below is the Quiz of previous year GATE Questions

https://www.geeksforgeeks.org/dbms-gq/database-design-normal-forms-gq/

In a database management system (DBMS), a lossless decomposition is a process of decomposing a relation schema into multiple relations in such a way that it preserves the information contained in the original relation. Specifically, a lossless decomposition is one in which the original relation can be reconstructed by joining the decomposed relations.

To achieve lossless decomposition, a set of conditions known as Armstrong’s axioms can be used. These conditions ensure that the decomposed relations will retain all the information present in the original relation. Specifically, the two most important axioms for lossless decomposition are the reflexivity and the decomposition axiom.

The reflexivity axiom states that if a set of attributes is a subset of another set of attributes, then the larger set of attributes can be inferred from the smaller set. The decomposition axiom states that if a relation R can be decomposed into two relations R1 and R2, then the original relation R can be reconstructed by taking the natural join of R1 and R2.

There are several algorithms available for performing lossless decomposition in DBMS, such as the BCNF (Boyce-Codd Normal Form) decomposition and the 3NF (Third Normal Form) decomposition. These algorithms use a set of rules to decompose a relation into multiple relations while ensuring that the original relation can be reconstructed without any loss of information.

In summary, a lossless decomposition is an important concept in DBMS that ensures that the original relation can be reconstructed from the decomposed relations without any loss of information. The use of Armstrong’s axioms and decomposition algorithms such as BCNF and 3NF can help achieve lossless decomposition in practice

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