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If a relation is in 3nf and atmost one compound key then?

1. Relation is in BCNF

2. Relation is not in BCNF.

3. Relation may be in BCNF

I was thinking like if i have a relation where R(ABCDE),where ABC is the candidate key(only one) and B->C exists in the FD set ,then this means B is not superkey so it cant be the BCNF,but answer given was that it will always be BCNF.Please tell what did i miss?

4 Answers

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1 votes
according to ur above question if a relation is given and  atmost one candidate key is given it means there would not be anoter fd in that set as well as no other candidate key , so it satisfy condition to fall in bcnf category
1 votes
1 votes
It's worth mentioning that a candidate key is a column or a group of column (one or more than one). So when they say that the relation R has 'at most one' candidate key, they mean the entire relation has just one column that define rows uniquely (primary key) or a group of columns (candidate keys).

Given that the above relation is in 3NF, and realizing that since there is just one candidate key, thus just one FD. Please look into the conditions of when a relation is in 3NF or BCNF.  

And so the above relation is always BCNF.
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0 votes
Candidate Key is minimal super keys.

Let's say we have a relation R(A, B, C) (short form of relation you have considered)

FD: B-> C

Candidate Key: AB ( Not ABC, as without C also we can find all other attributes).

Now check you will always see that if you have single CK, then 3NF table will also be in BCNF.
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relation to be in 3rd NF..

1.for all the FD's right side should be a prime attributes...

or

2.left side should be a key...(in this case relation is in BCNF )

let me explain..case 1.

if left side is not key and, right side is a prime attribute then ...if we take other prime attributes and concatenate  with left side of FD then it would form another candidate key...which is not acceptable corresponding to question....

ex:- R(ABCD)...AB is candidate key  FDs.{AB->CD ,C->B} 3rd NF but not BCNF

if this is the scenario then here AC also  would be candidate key...means two candidate key ...which is not allowed...

therefor relation would be in BCNF always....

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