**Joint dependency –** Join decomposition is a further generalization of Multivalued dependencies. If the join of R1 and R2 over C is equal to relation R then we can say that a join

dependency (JD) exists, where R1 and R2 are the decomposition R1(A, B, C) and R2(C, D) of a given relations R (A, B, C, D). Alternatively, R1 and R2 are a lossless decomposition of R. A JD ⋈ {R1, R2, …, Rn} is said to hold over a relation R if R1, R2, ….., Rn is a lossless-join decomposition. The *(A, B, C, D), (C, D) will be a JD of R if the join of join’s attribute is equal to

the relation R. Here, *(R1, R2, R3) is used to indicate that relation R1, R2, R3 and so on are a JD of R.

Let R is a relation schema R1, R2, R3……..Rn be the decomposition of R. r( R ) is said to satisfy join dependency if and only if

**Example –**

**Table –** R1

COMPANY |
PRODUCT |

C1 | pendrive |

C1 | mic |

C2 | speaker |

C2 | speaker |

Company->->Product

**Table –** R2

AGENT |
COMPANY |

Aman | C1 |

Aman | C2 |

Mohan | C1 |

Agent->->Company

**Table –** R3

AGENT |
PRODUCT |

Aman | pendrive |

Aman | mic |

Aman | speaker |

Mohan | speaker |

Agent->->Product

**Table –** R1⋈R2⋈R3

COMPANY |
PRODUCT |
AGENT |

C1 | pendrive | Aman |

C1 | mic | Aman |

C2 | speaker | speaker |

C1 | speaker | Aman |

Agent->->Product

**Fifth Normal Form / Projected Normal Form (5NF):**

A relation R is in 5NF if and only if every join dependency in R is implied by the candidate keys of R. A relation decomposed into two relations must have loss-less join Property, which ensures that no spurious or extra tuples are generated, when relations are reunited through a natural join.

**Properties –** A relation R is in 5NF if and only if it satisfies following conditions:

- R should be already in 4NF.
- It cannot be further non loss decomposed (join dependency)

**Example –** Consider the above schema, with a case as “if a company makes a product and an agent is an agent for that company, then he always sells that product for the company”. Under these circumstances, the ACP table is shown as:

**Table –** ACP

AGENT |
COMPANY |
PRODUCT |

A1 | PQR | Nut |

A1 | PQR | Bolt |

A1 | XYZ | Nut |

A1 | XYZ | Bolt |

A2 | PQR | Nut |

The relation ACP is again decompose into 3 relations. Now, the natural Join of all the three relations will be shown as:

**Table –** R1

AGENT |
COMPANY |

A1 | PQR |

A1 | XYZ |

A2 | PQR |

**Table –** R2

AGENT |
PRODUCT |

A1 | Nut |

A1 | Bolt |

A2 | Nut |

**Table –** R3

COMPANY |
PRODUCT |

PQR | Nut |

PQR | Bolt |

XYZ | Nut |

XYZ | Bolt |

Result of Natural Join of R1 and R3 over ‘Company’ and then Natural Join of R13 and R2 over ‘Agent’and ‘Product’ will be table **ACP**.

Hence, in this example, all the redundancies are eliminated, and the decomposition of ACP is a lossless join decomposition. Therefore, the relation is in 5NF as it does not violate the property of lossless join.

This helped me understand conceptually better than some other typed, but the examples have a lot of typos so it’s harder to follow along on them. If those get edited, I think this would be perfect.