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4 Expressing division and antidivision No specific feature is available in SQL to express division queries. Such a need is covered, thanks to partitioning. Recall that the division of relation r with schema R(A, X) by relation s with schema S(B, Y ) retains a value x appearing in attribute X of r if it is associated with at least all the values of attribute B in s. It turns out that partitioning enables the computation of the set of values associated with every x (using X as the partitioning attribute).

Disjunction); the n-ary version of min/max is taken later on (except otherwise stated), • the negation by the complement to 1, • membership (x ∈ E) by a degree of membership (µE (x)). In the specific case of the difference, letting E and F be two fuzzy sets, one obtains: ∀x ∈ X, µE−F (x) = (µE (x), µF (x)) = which, with particular choices for the norm (µE (x), 1 − µF (x)), , leads to: µE−F (x) = min(µE (x), 1 − µF (x)) with (x, y) = min(x, y), µE−F (x) = max(µE (x)−µF (x), 0) if (x, y) = max(x+y−1, 0) is chosen.

3/b}. 3), which highlights the variety of behaviors observed according to the choice of difference operator. FA January 3, 2012 15:34 Fuzzy Preference Queries To Relational Databases - 9in x 6in Basic Notions on Fuzzy Sets b1283-ch03 43 This kind of approach is particularly appropriate when it is impossible to call on the notion of complement, for instance if one wants to deal with fuzzy bags (see Rocacher and Bosc, 2002; Rocacher, 2002). A somewhat similar view is based on: E − F = the smallest set to add to F to obtain E ∪ F, which leads to: µE−F (x) = inf {u | ⊥(u, µF (x)) ≥ ⊥(µE (x), µF (x))}, where ⊥ is a triangular co-norm.

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