Dependency Preserving Decomposition in DBMS

Decomposition - This means replacing a relation with a collection of smaller relations. 

Relation - Relation is known as Table.

Relation Schema - This is known as Table definition. Relation Schema for a "student" relation can be shown in the following way:
Student(FirstName,LastName,DOB,Gender,Course,Regno,Address)

Definition of Dependency preservation decomposition:-
Each FD specified in F either appears directly in one of the relations in the decomposition, or be inferred from FDs that appear in some relation.

Let us consider an example for Dependency preservation

Let R be a relation R(A B C D)
Let there be 3 functional dependencies.
FD1: A->B
FD2: B->C
FD3: C->D
Let the relation R be decomposed into two more relations.
R1(A B C)  :  R2(C D)
Let us first consider the relation R1(A B C). Here between A and B the functional dependency FD1 is preserved. Between B and C, FD2 is preserved.
Let us now consider the second relation R2(C D). Between C and D the FD, FD3 is preserved. So in the two relations R1 and R2, all the 3 functional dependencies are preserved.

Let us consider an example for Non-dependency preservation

Let R be a relation R(A B C D) Let there be again 3 functional dependencies.
FD1:A->B
FD2:B->C
FD3:C->D
Let the relation be decomposed into two more relations>
R1(A C D) R2(B C)
Let us first consider the relation R1(A C D). There is no FD between A and C. There is a FD3 between C and D.
Now let us consider the second relation R2(B C). There is FD2 between B and C.
So, the two relations only support only FD's FD2 and FD3. FD1 is not supported. So these relations does not preserve dependency.
Generally there are three desirable properties of a decomposition.

1. Lossless
2. Dependency preservation
3. Minimal redundancy