# DBMS - Functional Dependencies

## Functional Dependencies

Functional dependencies are a way to describe the relationship between two or more attributes in a database table. Functional dependencies are denoted by the symbol $\to$ and are read as "if A then B". For example, if we have a table with the following attributes:

book_id | book_name | genre | author_id | author_name |
---|---|---|---|---|

1 | The Hobbit | Fantasy | 1 | J.R.R. Tolkien |

2 | Fire & Blood | Fantasy | 2 | George R.R. Martin |

3 | The Anthropocene Reviewed | Non-Fiction | 3 | John Green |

4 | A Game of Thrones | Fantasy | 2 | George R.R. Martin |

5 | The Fault in Our Stars | Fiction | 3 | John Green |

We can say that `book_id`

is functionally dependent on `book_name`

. This means that if we know the `book_id`

we can uniquely identify the `book_name`

. Similarly, we can say that `author_id`

is functionally dependent on `author_name`

. This means that if we know the `author_id`

we can uniquely identify the `author_name`

.

## Rules of Functional Dependencies

There are three rules that we need to follow when defining functional dependencies:

**Axiom of Reflexivity**: If B is a subset of A, then $A \to B$.**Axiom of Augmentation**: If $A \to B$, then $A + C \to B + C$.**Axiom of Transitivity**: If $A \to B$ and $B \to C$, then $A \to C$.

### Axiom of Reflexivity

The axiom of reflexivity states that if B is a subset of A, then $A \to B$. For example, a attribute `address`

, which contains `street`

, `city`

and `postal_code`

. We can say that ** address $\to$ {street, city, postal_code}**.

### Axiom of Augmentation

The axiom of augmentation states that if $A \to B$, then $A + C \to B + C$. Axiom of augmentation is also known as **partial dependency**.

### Axiom of Transitivity

The axiom of transitivity states that if $A \to B$ and $B \to C$, then $A \to C$. Axiom of transitivity is also known as **transitive dependency**.

## Types of Functional Dependencies

There are four types of functional dependencies:

- Trivial Functional Dependencies
- Non-Trivial Functional Dependencies
- Multivalued Dependencies
- Transitive Functional Dependencies

### Trivial Functional Dependencies

Trivial functional dependencies are those where the dependent attribute is a subset of the determinant attribute. In other words, if $A \to B$, then B is a subset of A. For example, if we have a table with the following attributes:

book_id | book_name | genre | author_id | author_name |
---|---|---|---|---|

1 | The Hobbit | Fantasy | 1 | J.R.R. Tolkien |

2 | Fire & Blood | Fantasy | 2 | George R.R. Martin |

3 | The Anthropocene Reviewed | Non-Fiction | 3 | John Green |

4 | A Game of Thrones | Fantasy | 2 | George R.R. Martin |

5 | The Fault in Our Stars | Fiction | 3 | John Green |

Here, if we take the FD ** {book_id, title} $\to$ title**, then

`title`

is a subset of `book_id`

and `title`

. This is a trivial functional dependency. We can also say that **is a trivial functional dependency.**

`book_id`

$\to$ `book_id`

### Non-Trivial Functional Dependencies

Non-trivial functional dependencies are those where the dependent attribute is not a subset of the determinant attribute. In other words, if **A -> B**, then B is not a subset of A. For example, if we have a table with the following attributes:

book_id | book_name | genre | author_id | author_name |
---|---|---|---|---|

1 | The Hobbit | Fantasy | 1 | J.R.R. Tolkien |

2 | Fire & Blood | Fantasy | 2 | George R.R. Martin |

3 | The Anthropocene Reviewed | Non-Fiction | 3 | John Green |

4 | A Game of Thrones | Fantasy | 2 | George R.R. Martin |

5 | The Fault in Our Stars | Fiction | 3 | John Green |

Here, if we take the FD ** {book_id, title} $\to$ genre**, then

`genre`

is not a subset of `book_id`

and `title`

. This is a non-trivial functional dependency.### Multivalued Dependencies

Multivalued dependencies are those where the determinant attributes are not dependent on each other. In other words, if **A $\to$ {B, C}**, then B and C are not dependent on each other. For example, if we have a table with the following attributes:

book_id | book_name | genre | author_id | author_name |
---|---|---|---|---|

1 | The Hobbit | Fantasy | 1 | J.R.R. Tolkien |

2 | Fire & Blood | Fantasy | 2 | George R.R. Martin |

3 | The Anthropocene Reviewed | Non-Fiction | 3 | John Green |

4 | A Game of Thrones | Fantasy | 2 | George R.R. Martin |

5 | The Fault in Our Stars | Fiction | 3 | John Green |

Here, if we take the FD ** book_id $\to$ {book_name, genre}**, then

`book_name`

and `genre`

are not dependent on each other. This is a multivalued dependency.### Transitive Functional Dependencies

Transitive functional dependencies are those where the determinant attributes are dependent on each other. In other words, if $A \to B$ and $B \to C$, then $A \to C$. For example, if we have a table with the following attributes:

book_id | book_name | genre | author_id | author_name |
---|---|---|---|---|

1 | The Hobbit | Fantasy | 1 | J.R.R. Tolkien |

2 | Fire & Blood | Fantasy | 2 | George R.R. Martin |

3 | The Anthropocene Reviewed | Non-Fiction | 3 | John Green |

4 | A Game of Thrones | Fantasy | 2 | George R.R. Martin |

5 | The Fault in Our Stars | Fiction | 3 | John Green |

Here, if we take the FD ** book_id $\to$ book_name** and

**, then**

`book_name`

$\to$ `genre`

**. This is a transitive functional dependency.**

`book_id`

$\to$ `genre`

## Key Points

- A functional dependency is a relationship between two or more attributes of a relation.
- A functional dependency is represented as $A \to B$, where A is the determinant attribute and B is the dependent attribute.
- Functional dependencies follow three rules:
**Axiom of Reflexivity**,**Axiom of Augmentation**and**Axiom of Transitivity**. - There are four types of functional dependencies: 1.
**Trivial Functional Dependencies**, 2.**Non-Trivial Functional Dependencies**, 3.**Multivalued Dependencies**and 4.**Transitive Functional Dependencies**.

## References

- Armstrong's axioms. Wikipedia. Retrieved October 29, 2022, from https://en.wikipedia.org/wiki/Armstrong%27s_axioms
- Functional dependency. Wikipedia. Retrieved October 29, 2022, from https://en.wikipedia.org/wiki/Functional_dependency
- GitHub Copilot · Your AI pair programmer · GitHub. GitHub. Retrieved October 29, 2022, from https://github.com/features/copilot
- Peterson, R. (2022, August 25). Functional Dependency in DBMS: What is, Types and Examples. Guru99. Retrieved October 29, 2022, from https://www.guru99.com/dbms-functional-dependency.html
- Types of Functional dependencies in DBMS. (2021, August 21). GeeksforGeeks. Retrieved October 29, 2022, from https://www.geeksforgeeks.org/types-of-functional-dependencies-in-dbms/