A trivial functional dependency is one which satisfies X Y. Can a key be the set of all attributes if there is at least one non-trivial functional dependency in a schema? Explain.

What will be an ideal response?

Suppose a key, k, contained all attributes. Suppose there is a non-trivial FD X?Y. Since k contains all attributes, it contains X and Y. But the FD says that X determines Y. Hence we can delete Y (X Y) from k and the remaining attributes still form a key. This means that k is not minimal, a contradiction.