Calculate the key for relation R in DBMS

Find the candidate keys of a relation, How to find the candidate keys, Which is the key for the given table, concept of candidate key in dbms, candidate key examples

Question:

Consider the relation R = {A, B, C, D, E, F, G, H, I, J} and the set of functional dependencies F as follows;

F = { AB → C, BD → EF, AD → GH, A → I, H → J}.

Find the key of relation R.

Solution:

We can find keys (candidate keys) for a relation by finding the closure of an/set of attributes. Checking each attribute or all subsets of the given set of attributes for a key is time consuming job. Hence, we may employ some of the following heuristics/assumptions in identifying the keys;

• We may start checking all the left hand side attributes of any/all of the given set of functional dependencies. We can start with single LHS attributes.

• If we find the closure of an attribute and that attribute is the candidate key then any superset cannot be the candidate key. For example, if A is a candidate key, then AB is not a candidate key but a super key.

LHS	Result	Description
A+	= AI from the functional dependency A → I (By reflexivity rule) No more functional dependencies that has either of the attributes AI in LHS. Hence, A+ = AI.	A+ ≠ R. So, A is not a candidate key.
H+	= HJ from H → J (By reflexivity rule) No more functional dependencies that has either of the attributes HJ in LHS. Hence, H+ = HJ. Note: We need not find H+ since H is available on RHS in AD → GH.	H+ ≠ R. So, H cannot be a candidate key.
(AB)+	= ABC from AB → C = ABCI from A → I No more functional dependencies that has either of the attributes ABCI in LHS. Hence, (AB)+ = ABCI.	(AB)+ ≠ R. So, (AB) is not a candidate key.
(AD)+	= ADGH from AD → GH (By reflexivity rule) = ADGHI from A → I (By union rule. AD → GH and A → I, so AD → GHI) = ADGHIJ from H → J No more functional dependencies that has either of the attributes ADGHIJ in LHS. Hence, (AD)+ = ADGHIJ.	(AD)+ ≠ R. So, (AD) is not a candidate key.
(BD)+	= BDEF from BD → EF No more functional dependencies that has either of the attributes BDEF in LHS. Hence, (BD)+ = BDEF.	(BD)+ ≠ R. So, (BD) is not a candidate key.
(ABD)+	= ABDGH (By reflexivity and augmentation. That is, AD → GH and if we augment B on both sides we get ABD → BGH. Hence, ABDGH.) = ABDGHIJ from (AD)+ - refer above. = ABCDGHIJ from AB → C = ABCDEFGHIJ from BD → EF.	(ABD)+ = R. Hence, (ABD) is the candidate key.

Hint: The left hand side only attribute will definitely be the key or part of the key.

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