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This lecture slide is very easy to understand and very helpful to built a concept about the foundation of computers and Database Design.The key points in these slides are:Sets of Attributes, Functional Dependency, Determinant Set, Dependent Attribute, Express Constraints, Closure of Relation, Set of Rules, Reflexivity Rule, Augmentation Rule, Transitivity Rule, Theorems from Axioms
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A functional dependency is defined as a constraint between two sets of attributes in a relation from a database.
Given a relation R , a set of attributes X in R is said to functionally determine another attribute Y , also in R , (written X → Y ) if and only if each X value is associated with at most one Y value.
X is the determinant set and Y is the dependent attribute. Thus, given a tuple and the values of the attributes in X , one can determine the corresponding value of the Y attribute.
Whereas a key is a set of attributes that uniquely identifies an entire tuple, a functional dependency allows us to express constraints that uniquely identify the values of certain attributes.
However, a candidate key is always a determinant, but a determinant doesn’t need to be a key.
Let a relation R have some functional dependencies F specified. The closure of F (usually written as F+ ) is the set of all functional dependencies that may be logically derived from F. Often F is the set of most obvious and important functional dependencies and F+ , the closure, is the set of all the functional dependencies including F and those that can be deduced from F. The closure is important and may, for example, be needed in finding one or more candidate keys of the relation.
Before we can determine the closure of the relation, Student, we need a set of rules.
Developed by Armstrong in 1974, there are six rules (axioms) that all possible functional dependencies may be derived from them.
1. Reflexivity Rule --- If X is a set of attributes and
each subset of X is functionally dependent on X.
SNo SName CNo CName Addr Instr. Office
Based on the rules provided, the following dependencies can be derived.
(SNo, CNo) → SNo (Rule 1) -- subset (SNo, CNo) → CNo (Rule 1) (SNo, CNo) → (SName, CName) (Rule 2) -- augmentation CNo → office (Rule 3) -- transitivity SNo → (SName, address) (Union Rule) etc.
Using the first rule alone, from our example we have 2^7 = 128 subsets. This will further lead to many more functional dependencies. This defeats the purpose of normalizing relations. So what now?
One way is to deal with one attribute or a set of attributes at a time and find its closure (i.e. all functional dependencies relating to them). The aim of this exercise is to find what attributes depend on a given set of attributes and therefore ought to be together. Docsity.com
Consider the following relation: student(SNo, SName, CNo, CName). We wish to determine the closure of (SNo, CNo). We have the following functional dependencies. SNo -> SName CNo -> CName
