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Fuzzy Tolerance Relation If the fuzzy relation R1 satisfies both reflexivity and symmetry then it is called ∼ as fuzzy tolerance relation R1 . A fuzzy tolerance relation can be reformed into ∼ fuzzy equivalence relation at most (n−1) compositions. This is given by: R1n−1 = R1 ◦ R1 ◦ · · · ◦ R1 = R . 8. 8 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 54 3 Classical and Fuzzy Relations is reflexive and symmetric. 6) . 5) . 5. The transitivity is satisfied hence equivalence relation is also satisfied. 9. Find whether the ⎡ 1 1 0 ⎢1 1 0 ⎢ 0 0 1 R=⎢ ⎢ ⎣0 0 0 0 1 0 given matrix is reflexive or not ⎤ 0 0 0 1⎥ ⎥ 0 0⎥ ⎥ by writing an M -file.
These two laws does not hold good for fuzzy sets. Since fuzzy sets can overlap, a set and its complement also can overlap. The excluded middle law for fuzzy sets is given by − A ∪ A = X, ∼ ∼ − A ∩ A = φ. ∼ ∼ Comparing Venn diagram for classical sets and fuzzy sets for excluded middle law are shown in Figs. 12. 2 Properties of Fuzzy Sets The properties of the classical set also suits for the properties of the fuzzy sets. The important properties of fuzzy set includes: Commutativity A ∪ B = B ∪ A, ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ A∩B = B ∩A.
The Cartesian product is denoted by A1 × A2 × · · · × An . In Cartesian product the first element in each pair is a member of x and the second element is a member of y formally, x × y = {(x, y)/x ∈ X and y ∈ Y }, if x = y then x × y = y × x. 38 3 Classical and Fuzzy Relations If all the An are identical and equal to A, then the cartesian product of A1 , A2 , . . , An becomes An . 1. The elements in two sets A and B are given as A = {0, 1} and B = {e, f, g} find the Cartesian product A × B, B × A, A × A, B × B.
Maple 11 Quick Reference Card
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