---

An algorithm for finding the closure set X⁺

---

• An Algorithm for finding X⁺
  • The following algorithm uses a number of inference rules to determined X⁺ for any set of attributes X:

    input: X (a set of attributes) ℉ = set of functional dependencies output: result = X+ result := X; (due to X → X) DONE := FALSE; while ( not DONE ) { DONE := TRUE; for ( each FD "Y → Z" ∈ ℉ ) do { if ( Y ⊆ result ) { if ( Z ⊄ result ) { result := result ∪ Z; DONE := FALSE; } } } }

    ---

  • NOTE: the algorithm works because:
    • Clearly: X → X⁺
    • If Y ⊆ X⁺, then we have X⁺ → Y
    • Because Y → Z, we also have X⁺ → Z
    • Because X → X⁺ we have X → Z
    • Then use the union rule to conclude that: X → X⁺ ∪ Z