NPTEL PPC DBMS lectures - Lecture 17 functional dependency 3

Question

The lecturer gives a set of funcional dependencies and the task is to get the canonical form.

The dependencies are as follows.

1.B->C

2.A->B and

3.AB->C

as proof for fd  fd 3 being redundant he tells that B+ = AB+. i.e closure of B and closure of AB are same. which i believe is INCORECT since B+ : {B,C} while AB+ : {A,B,C}.

I need someoene to verify this and clear my confusion.

Comments

You are correct.

$B^+ = \{ B,C\}$ only

Whereas $A^+ = \{A, B,C\}$

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Thank you so much for clarifying

Answer 1

Consider the functional dependency AB → C. If it contains an extraneous attribute, then either A or B may be extraneous. For instance, if A is extraneous, the dependency reduces to B → C, making A redundant. In such a case, the resulting set of functional dependencies would be A → B and B → C, and the original dependency AB → C would become redundant.

In general, for a functional dependency X A → Y, the attribute A is said to be extraneous if X → Y already holds.
Closure of B={B,C} .