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Finding the closure set X⁺ of a set of attribute X

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• The closure set X⁺ of a set of attribute X
  • Let X be some set of attributes of a R

    The closure set X⁺ is:

    set of attributes that are functionally determined by X

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  • Clearly:

    X ⊆ X+

    because X → X

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• The importance of the closure set X⁺
  • Property:
    Let R be a relation

    Let X be a set of attributes of R

    If X+ = R then X is a superkey

  • In other words:

    The closure can tell us if the set of attribute X is a super key

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  • How to proceed to find a key after computing   X⁺:

    If X+ ≠ R then: X is not a super key.... We must augment (= add) other attributes to X in order to obtain a key If X+ = R then: X is a super key However: X may not be a key because X is not minimal !!!! How to check for minimality: Remove one or more attributes A from X Compute the closure of X-A: If (X-A)+ = R then X is not minimal !!! (because you do not need A to be a super key !!!) X is a key if: For every attribute A:   (X−A)+ ≠ R            (Only then we know X is minimal)