Question

The following functional dependencies hold true for the relational schema R{V, W, X, Y, Z}:

V → W

VW → X

Y → VX

Y → Z

Which of the following is irreducible equivalent for this set of functional dependencies?
1.

V → W

V → X

Y → V

Y → Z
2.

V → W

W → X

Y → V

Y → Z
3.

V → W

V → X

Y → V

Y → X

Y → Z
4.

V → W

W → X

Y → V

Y → X

Y → Z

Answer 1

Correct Answer - Option 1 :

V → W

V → X

Y → V

Y → Z

Concept: 

To find the irreducible equivalent of the given functional dependencies, we need to find the minimal cover set for these functional dependencies by removing all the left extraneous attributes and extra dependencies.

Explanation

V → W

VW → X

Y → VX

Y → Z

Consider,

V → W (As this is a single dependency that must be required, no need to remove)

VW → X ( In this, we have to delete the left extra attribute, for this we need to find the V⁺  and W⁺ from original dependencies, Now doing V⁺ if in V⁺ we get a W, then we can remove W attribute otherwise not same in case of W⁺)

So here, V⁺ = {V W X} and W⁺ = {W}

So, W → X can be removed from these functional dependencies.

Y → VX (Y → X can be find out using Y → V, V → X)

So, the irreducible equivalent of the given functional dependencies is:

V → W

V → X

Y → V

Y → Z