Prove that a functional dependency satisfies the formal definition of multi-valued dependency.

Question

What will be an ideal response?

Answer 1

Suppose that a functional dependency X -> Y exists in a relation R={X, Y, Z}, and suppose
there are two tuples with the same value of X. Because of the functional dependency, they
must also have the same value of Y. Suppose the tuples are t 1 = < x, y, z 1 > and t 2 = < x,
y, z 2 >. Then, according to the definition of multivalued dependency, we must have two
tuples t 3 and t 4 (not necessarily distinct from t 1 and t 2 ) satisfying: t 3 [X]= t 4 [X]=
t 1 [X]= t 2 [X], t 3 [Y]= t 2 [Y], t 4 [Y]= t 1 [Y], t 3 [Z]= t 1 [Z], and t 4 [Z]= t 2 [Z]. Two tuples
satisfying this are t 2 (satisfies conditions for t 4 ) and t 1 (satisfies conditions for t 3 ).
Hence, whenever the condition for functional dependency holds, so does the condition for
multivalued dependency.