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Consider the following relation
A B C Tuple #
10 b1 C1 #1
10 b2 C2 #2
11 b4 C1 #3
12 b3 C4 #4
13 b1 C1 #5
14 b3 C4 #6

Given the previous state which of the following dependencies may hold in the above relation?

If the dependency cannot hold explain why by specifying the tuples that cause the violation:-

I. A → B

II. B → C

III. C → B

IV. B → A

V. C → A

1 Answer
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What does X → Y mean? It means, Y is functionally dependent on X.

Remember this trick “Keep Y value same and check if X is same too. If not, relation doesn’t hold.”

I. A → B

This dependency cannot hold, because A → B implies that A can uniquely determine B.

However this is true not as seen in the above relation –

10 → b1#1

13 → b1 #5

..Which contradicts the dependency. Hence A→B doesn’t hold.

II. B → C

This dependency doesn’t holds, because C cannot be uniquely determined by B alone.

b1 → C1 #1

b4 → C1#3 contradicts

III. C → B

Holds good.

C1 → b1#1 #5

C4 →b3 #4 #6

Hence every value of B can be uniquely identified by C.

IV. B → A

Doesn’t hold. (Now you must reason this out.)

b1 → 10 #1

b1 → 10 #2 contradicts

V. C → A

Doesn’t hold.

C1 → 10 #1

C2 → 10 #2 contradicts

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