Keys in a Decomposition

Question

Relation $R=(L, M, N, O, P)$

$R$ is decomposed into $R_{1} = (L, M, N, P)$ and $R_{2} = (M, O).$

for $R_{1}$ dependencies are

$P->L, L->MN$

$1)$What are keys for $R_{1}$?

$2)$What normal form it holds?

Answer 1

Key for R1 = P

as P - > { P, L, M, N }

It's in 2nd Normal Form, as we don't have Partial Dependency but we have Transitive Dependency.

Comments

not two, with 4 dependencies. as mentioned in https://gateoverflow.in/869/gate2002-16

M→O, 

NO→P,

P→L

and 

L→MN

If we have all 4 dependencies, then we'll have BCNF.

For R1, as you can see all the attributes are Prime Attributes as all of them are part of Candidate Key.

For R2, we have M->O and M is the Key here, every binary relation with a FD in which LHS is a key is in BCNF.

---

for R1 According to me

MN is primary key

$(MN)^{+}=MNOPL$

how others will be key for R_{1}??

Am I mistaking anything?

---

for R1,

$(L)^+ = {L,M,N,P}$

L -> MN

M->O, NO->P implies MN->P (By Pseudo Transitivity Rule)

and

$(P)^+ = {L,M,N,P}$

P-> L and L-> MNP

I hope it's clear now.

As you can see all the attributes of R1 are Prime Attributes as they belong to Key.

Therefore, R1 is in BCNF

Answer 2

R1 is on 2NF.

Comments

no here keys are P,L,MN

and it is in BCNF

see it https://gateoverflow.in/869/gate2002-16

rt?

Answer 3

CK= {P}  and Relation is in 2NF.