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## Lossless Decomposition

Let r(R) be a relation schema.

Let R be decomposed into R1 and R2. The corresponding instances of R1 and R2 are r1 and r2 respectively.

Here, the decomposition of R into R1 and R2 is said to be ‘Lossless decomposition’, if

πR1 (r )  |X|  πR2 (r)  =  r .

Note :

Let  R1 and R2 be the decomposition of R, F be a set of FDs on R.

Here, this decomposition of R into R1 and R2 holds good, if at least one of the following satisfies.

R1 ∩ R2  →  R1  (or) R1 ∩ R2  →  R2

Example-1:

Let R = (A, B, C, D, E),  R1 = (A, B, C),  R2 = (C, D, E)

r :

 A B C D E 5 8 11 13 15 7 9 13 14 16 8 10 14 15 17

Prove that the decomposition of R in R1 and R2 is lossless.

Output :

 R1 R2 A B C C D E 5 8 11 11 13 15 7 9 13 13 14 16 8 10 14 14 15 17

Here,  R = R1  U  R2.

Here,   C is a common attribute. And

Let   r ’ =  r1 |X| r2 is as follows :

 A B C D E 5 8 11 13 15 7 9 13 14 16 8 10 14 15 17

Hence proved,  r   =   r ‘

Therefore, this is lossless decomposition.

Reference

Lossless Decomposition