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Functional Dependency and Attribute Closure in DBMS

In this tutorial, we will learn about functional dependency and attributes closure in the database management system. By Prerana Jain Last updated : May 27, 2023

Functional Dependency

A relational Database management System (RDBMS) represents the database o a collection of relations/tables. A functional dependency is a constraint between two sets of attributes in a relation. It is the property of semantics or meaning of attribute. Functional dependency is also a property of relational schema "R" and not a particular legal relation "r" of "R".

Now let us consider, a relational "a" schema "R" and let "x" and "y" be the two set of attributes, now there is a functional dependency from "x" to "y"

If, t1[x] = t2[x] then, t1[y] = t2[y]

Here, "x" is the determinant and "y" is the dependent or "x" determines "y".

  • Some functional dependency is directly visible in relational model but some either set of dependencies also hold good which are not directly visible.
  • The entire set of functional dependency is called as complete set.
  • Therefore, we must know how to calculate closure set of functional dependencies before Normalization.
  • A functional dependency from "A" to "B" is said to be trivial if "B" is a subset of "A".


In the corresponding relation,

Tuple   A       C
t1      a1      c1
t2      a1      c1
t3      a2      c2
t3      a2      c2
t4      a3      c3
t5      a3      c2

A → C holds as
t1[A] = t2[A] then t1[C] = t2[C]
t3[A] = t3[A] then t3[C] = t4[C]
but, C → A does not holds as.
t3[C] = t4[C] = t3[C] but t3[A] = t4[A] does not equal t5[A].

How to find whether a functional dependency is valid or invalid?

  • Step 1: Check if all the values of "A" are distinct than it is valid otherwise invalid.
  • Step 2: Check if all the values of the "B" are same than the functional dependency is also valid.
  • Step 3: Otherwise we have to find at least one value of "A" on which are having different values of "B" than the functional dependency is invalid.

Attribute Closure

The set "A*" is said to be the closure set of "A" if the set of attributes are functionally dependent on the attributes of "A"

Inference rules to calculate the closure set

Some inference rules to calculate the closure set:

  1. Reflexive rule: A rule is said to be reflexive if B is a subset of a then A → B. This is called trivial functional dependency rule.
  2. Augmentation rule: A rule is said to be augmented if A → B then Ar → Br holds good.
  3. Transitive rule: A rule is said to be transitive if A → B, B → r then A → r.
  4. Union rule: A rule is said to be union if A → B, B → r then A → Br also holds good.
    • Decomposition rule: A rule is said to be decomposed if A → Br, A → B then A → r.
    • Pseudo transitivity rule: If A → B and BW → r then WA → r.

First three rules are functionally complete and known as RAT rule or RAT axioms and also called Armstrong rule which means only theses rules are sufficient enough to find closure set.

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