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List all functional dependencies satisfied by the relation.
a b C
A1 B1 C1
A1 B1 C2
A2 B1 C1
A2 B1 C3
1 Answer
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1. The functional dependencies satisfied by the relation are as follows:

i. A → B

For the tuples t1(A1,B1,C1) and t2(A1,B1,C2), t1 (A) =t2(A)=A1 and t1(B)=t2(B)=B1. Thus, the functional dependency A→B is satisfied by the relation.

ii. C → B

For the tuples t1(A1,B1,C1) and t3(A2,B1,C2), t1 (C) =t3(C) =C1 and t1 (B)=t3(B)=B1. Thus, the functional dependency C→B is satisfied by the relation.

iii. The dependency AC→B can also be obtained.

2. Following are the functional dependencies which are not satisfied.

i. C does not functionally determine A because the tuples t1(A1,B1,C1) and t3(A2,B1,C1) have the same C but different A values.

ii. The same tuples also show B does not functionally determine A.

iii. A does not functionally determine C because the first two tuples t1(A1,B1,C1) and t2(A1,B1,C2)have the same A value and different C values.

iv. The same tuples also show B does not functionally determine C.

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