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databasefunctional-dependencies

SQL Functional dependencies

Suppose X1 → Y1 and X2 → Y2

Is it true that X1 ∩ X2 → Y1 ∩ Y2? How about X1 ∪ X2 → Y1 ∩ Y2?

I've been thinking about this for a couple of hours and am really stuck. Maybe the second one is true because anything both in Y1 and Y2 will be dependent on at least one of X1 or X2.

Solution

  • The first formula is obviously false. A very simple example to show this is:

    R(A,B,C,D)
A B → C D
B E → D F

    from this one cannot infer that B → D in any way, and in fact the following instance satisfies the two above dependencies, but not the third one (for the same value of B, there are two different values of D):

    A   B   C   D   E   F
----------------------
a1  b1  c1  d1  e1  f1
a2  b1  c1  d2  e1  f1

    The second formula is, on the other hand, true, and this can be proved by using the Armstrong’s Axioms.