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Chapter 5: Bipolar Fuzzy Sets and Equili... > FUZZY EQUILIBRIUM RELATIONS

FUZZY EQUILIBRIUM RELATIONS

Bipolarity and fuzziness opened a door for the generalization of Zadeh’s similarity relation (Zadeh, 1971) to nonlinear bipolar fuzzy equilibrium relation (Zhang, 2006). We extend the notion of L-fuzzy equilibrium relation (Zhang, 2005a) to include strong reflexivity, weak reflexivity, and non-reflexivity as in the follows.

  • Definition 5.4. A (binary) bipolar fuzzy relation R in X to BF, where X = {xi}, 1<i≤n, is bipolar symmetric if, ∀i,k, 0<i,k≤n, we have μR(xi,xk) = μR(xk,xi); (5.15)

it is positive pole reflexive if, ∀i, 0<i≤n, we have

μR(xi,xi) = (x,1); (5.16a)

it is negative pole reflexive if, ∀i, 0<i≤n, we have

μR(xi,xi) = (-1,y); (5.16b)

it is bipolar reflexive if, ∀i, 0<i≤n, we have


  

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