Find minimal cover of set of functional dependencies example, Solved exercise - how to find minimal cover of F? Easy steps to find minimal cover of FDs, What is minimal cover?
Question:
5. Find the minimal cover of the set of functional dependencies given; {A → BC, B → C, AB → D}
Solution:
Minimal cover:
Definition 1:
A minimal cover
of a set of FDs F is a minimal set of functional dependencies Fmin
that is equivalent to F. There can be many such minimal covers for a set of functional
dependencies F.
Definition 2:
A set of FDs F is
minimum if F has as few FDs as any equivalent set of FDs.
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Simple
properties/steps of minimal cover:
1. Right Hand Side (RHS) of all FDs
should be single attribute.
2. Remove extraneous attributes. [What is extraneous attribute? Refer here].
3. Eliminate redundant functional
dependencies.
Let us apply these properties to F = {A
→ BC, B → C, AB → D}
1. Right
Hand Side (RHS) of all FDs should be single attribute. So we write F as
F1, as follows;
F1 = {A → B, A → C, B → C, AB → D}
2. Remove
extraneous attributes.
Extraneous attribute is a redundant
attribute on the LHS of the functional dependency. In the set of FDs, on AB → D
has more than one attribute in the LHS. Hence, we check one of A and B is
extraneous or not.
First we check whether A is extraneous or not. To do that, we need to find the
closure of the remaining attribute B with respect to F1.
B+ = BC.
This does not include D, so A is not
extraneous.
Now we check whether B is extraneous or not. To do that, we need to find the
closure of the remaining attribute A with respect to F1.
A+ = ABCD.
This includes D, so B is extraneous,
ie., we can identify D without B on the LHS.
Now, we can write the new set of FDs,
F2 as follows;
F2 = {A → B, A → C, B → C, A → D}
3. Eliminate
redundant functional dependency.
If A → B, and B → C, then A → C is
true (according to transitive rule). Hence, the FD A → C is redundant. We can
eliminate this and we get final set of FDs F3 as follows;
F3 = {A → B, B → C, A → D}
The set of FDs F3 is the minimal cover of F.
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Similar topics
How to find closure of set of functional dependencies?
How to find closure of attributes?
How to find canonical cover for a set of functional dependencies?
How to find extraneous attribute?
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