A Functional Dependency (FD) X → Y means:
If two tuples have the same X value, they must have the same Y value.

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Armstrong's Axioms

1. Reflexivity: If Y ⊆ X, then X → Y

2. Augmentation: If X → Y, then XZ → YZ

3. Transitivity: If X → Y and Y → Z, then X → Z

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Closure of Functional Dependencies (F⁺)

All FDs that can be derived from a given set F.

Example:
F = {A → B, B → C}
F⁺ = {A → B, B → C, A → C, A → A, B → B, C → C, …}

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Closure of Attribute Sets (X⁺)

All attributes functionally determined by set X.

Algorithm:

1. Initialize X⁺ = X

2. For each Y → Z in F: if Y ⊆ X⁺, add Z to X⁺

3. Repeat until no changes

Example:
F = {A → B, B → C}
A⁺ = {A, B, C} (A determines B, B determines C)

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Canonical Cover (Minimal Cover)

Simplified equivalent set of FDs with:

1. Single attributes on the right side

2. No extraneous attributes on the left

3. No redundant FDs

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Additional Dependency Types

1. Trivial Dependency

• A dependency X → Y is trivial if Y ⊆ X.

• Always true, adds no new information.

Example:

• {A, B} → A is trivial (because A is part of {A, B}).

• {RollNo, Name} → RollNo

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2. Partial Dependency

• A partial dependency occurs when a non-prime attribute (not part of any candidate key) is functionally dependent on part of a composite candidate key, not the whole key.

• Problematic because it violates 2NF.

Example:
Relation: R(StudentID, CourseID, Grade, StudentName)

• Candidate Key = {StudentID, CourseID}

• FD: StudentID → StudentName
  👉 Here, StudentName depends only on part of the key (StudentID), not the whole {StudentID, CourseID}.
  So, partial dependency exists.

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3. Transitive Dependency

• A transitive dependency occurs when a non-prime attribute depends on another non-prime attribute via a candidate key.

• Problematic because it violates 3NF.

Example:
Relation: R(StudentID, DeptID, DeptName)

• Candidate Key = {StudentID}

• FDs: StudentID → DeptID, DeptID → DeptName
  👉 StudentID → DeptName is implied, but DeptName depends transitively on StudentID through DeptID.
  So, transitive dependency exists.

A transitive dependency exists when:

• X → Y (X is a candidate key or contains a candidate key),

• Y → Z,

• Y is not a candidate key, and

• Z is a non-prime attribute.

So the problem occurs if a non-prime attribute (Z) is indirectly dependent on a key through another non-prime attribute (Y).