Question

Show that every schema consisting of exactly two attributes must be in BCNF regardless of the given set F of functional dependencies.

Answer 1

Solution:

Explanation:

BCNF:

=>A relation is called in BCNF if every functional dependency of relation is of type X -> Y where X is super key.

Proving relation with exactly two attributes in BCNF:

=>Let say two attribute are A and B so there can be 4 cases of functional dependencies.

(i) Only A -> B functional dependency is present.

=>If A -> B functional dependency then A is super key because it can derive both the attributes A and B hence it is satisfying BCNF.

(ii) Only B -> A functional dependency then B is super key because it can derive both the attributes A and B hence it is satisfying BCNF.

(iii) There is no functional dependency in relation R(A, B).

=>If there is no functional dependency then relation is in BCNF because it is not voilating functional dependency rule.

(iv) Both non trivial functional dependencies A -> B and B -> A are present in the relation R(A, B) then A and B are superkeys hence is it satisfying BCNF.

=>Hence on the basis of above statements we can say that relation with exactly 2 attribute is always in BCNF.

I have explained each and every part with the help of statements attached to it.