How can I prove / disprove If A ↠ BC and A → B then A → C

Question

/ From GATE 2007 IT:

Consider the following implications relating to functional and multi valued dependencies given below, which may or may not be correct.

A) If A ↠ B and A ↠ C then A → BC
B) If A → B and A → C then A ↠ BC
C) If A ↠ BC and A → B then A → C
D) If A → BC and A → B then A ↠ C

Exactly how many of the above implications are valid?

A) 0 B) 1 C) 2 D) 3

I've just studied 4th Normal Form & Multi Valued Dependencies & I'm struggling to apply inference rules for multi Valued dependencies.

I've searched online though I've not found any resource on it.

I tried:

A) If A ↠ B and A ↠ C then A → BC. This is incorrect. It is very obvious to see.

B) If A → B and A → C then A ↠ BC. This can be proved by using Augmentation rule of FD. A ->B & A-> C. Then we get A->BC. Using Replication rule we get ↠ BC.

D) If A → BC and A → B then A ↠ C. This is true, because A->BC. From which we can get A->C. From which using Replication rule we can get A ↠ C.

C) If A ↠ BC and A → B then A → C. Here using Coalescence rule I'm getting A->B back. I'm not able to prove/disprove A → C.

1. Please help me prove/disprove option C.

2. What does it means to have a multi valued dependency on two attributes like A ↠ BC? All examples I've seen in books are like Employee ↠ Project | Dependent, where Employee simultaneously determines two attributes. So we end up storing duplicated. But I have never encountered a multi valued dependency with two attributes on the right hand side. Please give an example of such a dependency.

Answer 1

A,B and C are sets of attributes.

All examples I've seen in books are like Employee ↠ Project | Dependent, where Employee simultaneously determines two attributes.

A ↠ B | C is a special notation different from A ↠ BC. It's useful because MVDs always come in pairs. It says that both A ↠ B and A ↠ C hold. This implies that the relation equals AB JOIN AC. Ie the relation satisfies join dependency *{AB,AC}.

2. What does it means to have a multi valued dependency on two attributes like A ↠ BC?

Please give an example of such a dependency.

Just think of values for subrows the way you would think of values for one attribute. Suppose {W} ↠ {X}. Then whenever W appears it's with all of some set of X values. Suppose each X has a numerator N and denominator D. Then we could have a relation {W} ↠ {N,D}. Whenever W appears it's with all of some set of < N, D > subrow values.

1. Please help me prove/disprove option C.

C) If A ↠ BC and A → B then A → C. Here using Coalescence rule I'm getting A->B back. I'm not able to prove/disprove A → C.

Hint: In my last paragraph if {W} → {N} then the variation must come from D.

Hint: If you are permitted to prove by counterexample then it's good to try to disprove things via guessing at one or more trivial examples. Eg with {W} ↠ {N,D} and {W} → {N} (A={W}, B={N}, C={D}):

  V   W   N   D
=================
| v | w | n | 1 |
-----------------
| v | w | n | 2 |
-----------------

( How do I apply this definition of multivalued dependency (MVD) to this example relation? )