R = (J,K,L,M,N)
with a set of functional dependencies {J->KL,LM->N,K->M,N->J}
.
I understand the definition of BCNF. I believe that there exists no trivial functional dependencies and there may not be a super key. I'm not sure about the second part. How would you determine a super key from letters? Would appreciate some input on this.
The relation would be in Boyce-Codd Normal Form (BCNF), if the closure of the left-side attributes for all functional dependencies contains all relation attributes (J, K, L, M, N)
. In other words, the left-side attributes of each functional dependency contain a key.
Let's analyze your functional dependencies:
J -> KL
. Then K -> M
, then LM -> N
and N -> J
. So, J -> KL
satisfies BCNF. LM -> N
. Then N -> J
, then J -> KL
and that is all, we have all attributes. K -> M
. This functional dependency is obviously a violation of BCNF, because we cannot get more attributes from set of dependencies. N -> J
. Then J -> KL
and K -> M
. It satisfies BCNF. So, the third dependency violates BCNF and K
attribute is not the key itself.
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