Given the following relation instance R(x, y, z): x y z 1 4 2 1 5 3 1 6 3 3 2…
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Given the following relation instance R(x, y, z):
x y z
1 4 2
1 5 3
1 6 3
3 2 2
Which of the following functional dependencies are satisfied by this instance?
- A.
XY -> Z and Z -> Y
- B.
YZ -> X and Y -> Z
- C.
YZ -> X and X -> Z
- D.
XZ -> Y and Y -> X
Attempted by 85 students.
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Correct answer: B
The correct answer is: YZ -> X and Y -> Z.
A functional dependency A -> B is satisfied by an instance if every pair of rows with the same A value also has the same B value.
Check each relevant dependency:
XY -> Z holds because all (X, Y) pairs are distinct. However, Z -> Y fails because Z = 2 occurs with Y = 4 and Y = 2.
YZ -> X holds because all (Y, Z) pairs are distinct. Y -> Z also holds because every Y value in the instance is distinct.
X -> Z fails because X = 1 appears with Z = 2 and Z = 3.
XZ -> Y fails because (X, Z) = (1, 3) occurs with two different Y values, 5 and 6. Y -> X holds, but both dependencies in an option must hold.
Therefore, only option B has both functional dependencies satisfied by the instance.