# Functional Dependencies, Closures and Keys — Database Fundamentals

Source: https://www.skillbyai.com/en/database-fundamentals/n-fd

> Reason with functional dependencies, compute attribute closures and find candidate keys.

## Rules hidden in the data

A **functional dependency (FD)** X → Y holds in a relation if any two tuples that agree on attributes X also agree on Y: `roll_no → name` says one roll number determines exactly one name. FDs come from the meaning of the data, not from one sample. **Armstrong's axioms** derive new FDs: **reflexivity** (if Y ⊆ X then X → Y), **augmentation** (if X → Y then XZ → YZ) and **transitivity** (if X → Y and Y → Z then X → Z); useful derived rules are **union**, **decomposition** and **pseudo-transitivity**. The **closure** of an attribute set, X⁺, is every attribute functionally determined by X: start with X and repeatedly add the right side of any FD whose left side is already included. X is a **super key** if X⁺ contains all attributes, and a **candidate key** if additionally no proper subset of X is a super key. A **minimal (canonical) cover** is a simplified equivalent set of FDs with single attributes on the right, no extraneous attributes and no redundant FDs.

## Computing an attribute closure

Start from a set of attributes and keep adding everything the dependencies let you reach.

![A small seed circle expanding outward in rings, each ring adding new attribute dots until the whole set is covered.](assets/figures/database-fundamentals/section-4-map.svg) — Figure 4.1 — Growing X⁺ by applying functional dependencies.

## Closure and candidate keys, worked by hand

R(A, B, C, D, E) with F = { A → B, B → C, CD → E }

```text
compute {A, D}+ :
  start          {A, D}
  A -> B         {A, B, D}
  B -> C         {A, B, C, D}
  CD -> E        {A, B, C, D, E}   all attributes -> AD is a super key

is AD minimal?
  {A}+ = {A, B, C}         not all attributes
  {D}+ = {D}               not all attributes
  -> AD is a candidate key

A and D never appear on the right side of any FD, so every key must contain both;
hence AD is the only candidate key.
```

## Attributes never on a right side are in every key

When finding candidate keys, first collect attributes that never appear on the right-hand side of any FD; they must be part of every key. Then add attributes only as needed until the closure covers everything.

**Quiz:** Given R(A, B, C) and F = { A → B, B → C }, what is {A}⁺?

- [ ] {A}
- [ ] {A, B}
- [x] {A, B, C}
- [ ] {B, C}

*Answer:* {A, B, C}. A determines B, and B determines C, so the closure of A contains all three attributes.