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Which one of the following statements are not correct?

$S1$: $3$NF decomposition is always lossless join and dependency preserving.

$S2$: $3$NF decomposition is always lossless join but may or may not be dependency preserving.

$S3$: BCNF decomposition is always lossless join and dependency preserving.

$S4$: BCNF decomposition is always lossless join but may or may not be dependency preserving.

1. Only $S1$
2. Only $S4$
3. Both $S1$ and $S4$
4. Both $S2$ and $S3$

### 1 comment

option D

Both S2 and S3
There will always be a decomposition to 3NF which is lossless and dependency preserving, For BCNF one such decomposition will always be there which is lossless but dependency preserving isn't guaranteed.

So S2 and S3 is wrong. Hence D is correct.
by
→ Sometimes, BCNF may not be functional dependency preserving.
→ lossless join should be compulsory in any normal form.
by

$\text{3NF vs BCNF:}$
Consider the functional dependency$: A \rightarrow B$
Condition for $\text{3NF:}$
The $A$ is Super key (or) $B$ is a prime attribute.
Condition for $\text{BCNF:}$
The $A$ should be Super key whether $B$ is prime or non-prime attribute.

• $\text{3NF}$ decomposition is always lossless join and dependency preserving.
• $\text{BCNF}$ decomposition is always lossless join but may or may not be dependency preserving.

So, the correct answer is $(D).$