Given an instance of the relation R(ABCD). A B C D a1 b1 c1 d1 a1 b2 c2 d2 a2 b2 c2 d3 a3 b3 c4 d3   Which of the following functional dependencies hold?

Given an instance of the relation R(ABCD). A B C D a1 b1 c1 d1 a1 b2 c2 d2 a2 b2 c2 d3 a3 b3 c4 d3   Which of the following functional dependencies hold? Correct Answer {AB} → C and B → C

Concept:

Rules for defining functional dependencies(FD) :

1) If LHS of FD are all different, then we can directly say that FD holds true.

2) If LHS of FD are same, then check the corresponding value of RHS, that should be same for all the matching value of LHS. Otherwise FD will not hold true.

Explanation:

Option 1: {AB} → D and D → A

Here, first consider AB -> D

Check AB first, all the values of AB are different, so AB-> D holds true.

Now check D-> A, in D two values are same i.e. d₃. So, we have to check the corresponding RHS of d₃.

These are different. As for one d₃, value of A is a₂ and for another d₃ value of A is a₃. So, D-> A does not hold true.

Option 2: AB-> C and B-> D

AB -> C holds true. As all the values of AB are different.

For B-> D, first check LHS, all the values of B are not same. There are two b₂ in B.

So, check RHS for b₂ in D. Both values are different : one is d₂ and another is d_3­. So, B -> D does not hold true.

Option 3: AB -> C and B-> C

AB -> C holds true. As all the values of AB are different.

Now check B-> C, in B all the values are not same. So, we have to check value of C corresponds to b₂. Both the values in C are same for b₂. So, B -> C holds true.

So, in this both the FD holds true.

Option 4: AB -> D and A -> D

AB -> D holds true. Check A -> D. All the values are not same in A i.e. a₁. SO check the values in D for a₁ Values in D for a₁ are different one is d₁ and another is d₂. So, A-> D does not hold true.