What does it mean to say that one attribute of a table is functionally dependent on another?

Prerequisite: Functional dependency and attribute closure

A functional dependency is a constraint that specifies the relationship between two sets of attributes where one set can accurately determine the value of other sets. It is denoted as X → Y, where X is a set of attributes that is capable of determining the value of Y. The attribute set on the left side of the arrow, X is called Determinant, while on the right side, Y is called the Dependent. Functional dependencies are used to mathematically express relations among database entities and are very important to understand advanced concepts in Relational Database System and understanding problems in competitive exams like Gate.

Example:

From the above table we can conclude some valid functional dependencies:

• roll_no → { name, dept_name, dept_building },→  Here, roll_no can determine values of fields name, dept_name and dept_building, hence a valid Functional dependency
• roll_no → dept_name , Since, roll_no can determine whole set of {name, dept_name, dept_building}, it can determine its subset dept_name also.
• dept_name → dept_building ,  Dept_name can identify the dept_building accurately, since departments with different dept_name will also have a different dept_building
• More valid functional dependencies: roll_no → name, {roll_no, name} ⇢ {dept_name, dept_building}, etc.

Here are some invalid functional dependencies:

• name → dept_name   Students with the same name can have different dept_name, hence this is not a valid functional dependency.
• dept_building → dept_name    There can be multiple departments in the same building, For example, in the above table departments ME and EC are in the same building B2, hence dept_building → dept_name is an invalid functional dependency.
• More invalid functional dependencies: name → roll_no, {name, dept_name} → roll_no, dept_building → roll_no, etc.

Armstrong’s axioms/properties of functional dependencies:

1. Reflexivity: If Y is a subset of X, then X→Y holds by reflexivity rule
   For example, {roll_no, name} → name is valid.
2. Augmentation: If X → Y is a valid dependency, then XZ → YZ is also valid by the augmentation rule.
   For example, If {roll_no, name} → dept_building is valid, hence {roll_no, name, dept_name} → {dept_building, dept_name} is also valid.→
3. Transitivity: If X → Y and Y → Z are both valid dependencies, then X→Z is also valid by the Transitivity rule.
   For example, roll_no → dept_name & dept_name → dept_building, then roll_no → dept_building is also valid.

Types of Functional dependencies in DBMS:

1. Trivial functional dependency
2. Non-Trivial functional dependency
3. Multivalued functional dependency
4. Transitive functional dependency

1. Trivial Functional Dependency

In Trivial Functional Dependency, a dependent is always a subset of the determinant.
i.e. If X → Y and Y is the subset of X, then it is called trivial functional dependency

For example,

Here, {roll_no, name} → name is a trivial functional dependency, since the dependent name is a subset of determinant set {roll_no, name}
Similarly, roll_no → roll_no is also an example of trivial functional dependency. 

2. Non-trivial Functional Dependency

In Non-trivial functional dependency, the dependent is strictly not a subset of the determinant.
i.e. If X → Y and Y is not a subset of X, then it is called Non-trivial functional dependency.

For example,

Here, roll_no → name is a non-trivial functional dependency, since the dependent name is not a subset of determinant roll_no
Similarly, {roll_no, name} → age is also a non-trivial functional dependency, since age is not a subset of {roll_no, name} 

3. Multivalued Functional Dependency

In Multivalued functional dependency, entities of the dependent set are not dependent on each other.
i.e. If a → {b, c} and there exists no functional dependency between b and c, then it is called a multivalued functional dependency.

For example,

Here, roll_no → {name, age} is a multivalued functional dependency, since the dependents name & age are not dependent on each other(i.e. name → age or age → name doesn’t exist !)

4. Transitive Functional Dependency

In transitive functional dependency, dependent is indirectly dependent on determinant.
i.e. If a → b & b → c, then according to axiom of transitivity, a → c. This is a transitive functional dependency  

For example,

Here, enrol_no → dept and dept → building_no, 
Hence, according to the axiom of transitivity, enrol_no → building_no is a valid functional dependency. This is an indirect functional dependency, hence called Transitive functional dependency.

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