Consider a relation R=(M, N, O, P, Q) with the dependencies: {M,N ->O; O,P->Q ; P, Q-> N}. The key for relation R is :a.M, N, Ob.M, N, Pc.N, O, Pd.M, N
Question
Consider a relation R=(M, N, O, P, Q)
with the dependencies:
{M,N -> O; O,P -> Q; P, Q -> N}.
The key for relation R is:
- a. M, N
- b. N, O
- c. N, O
- d. M, N
Solution
To find the key for the relation R, we need to find the minimal set of attributes that can determine all other attributes in the relation.
Step 1: Find the closure of each attribute set on the left side of the functional dependencies.
- Closure of {M, N} is {M, N, O, Q} (using M,N ->O and O,P->Q)
- Closure of {O, P} is {O, P, N, Q} (using O,P->Q and P, Q-> N)
- Closure of {P, Q} is {P, Q, N} (using P, Q-> N)
Step 2: Check which closure set includes all attributes of the relation.
- The closure of {M, N} does not include all attributes of the relation R. It is missing P.
- The closure of {O, P} includes all attributes of the relation R.
- The closure of {P, Q} does not include all attributes of the relation R. It is missing M and O.
So, the key for the relation R is {O, P}, which corresponds to option c. N, O, P.
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