- A binary fuzzy relation R is represented by x R y where (x, y) ∈ R(X, Y). In-detail it will be represented by α / xRy where µR(x, y) = α
- The domain of a fuzzy relation R(X, Y) is the fuzzy set represented by dom R(X, Y) whose membership function is defined as
- The Range of a fuzzy relation R(X, Y) is a fuzzy set whose membership function is defined as
Therefore the strength of the strongest relation that each element of Y has to a element of X is equal to the membership value of that element in the range of R.
- Height of a fuzzy relation R is a number h(R) defined by
This height is nothing but the largest membership value attained by any pair (x, y) in R. If h(R) = 1 then the relation is normal otherwise it is called sub-normal.
Fuzzy Tolerance and Equivalence Relation
A fuzzy relation ‘R’ is a relation from R : x to X then,
- Reflexivity µR(xi, xi) = 1
- Symmetry µR(xi, xj) = µR(xj, xi)
- Transitivity µR(xi, xj) = λ1 and µR(xj, xk) = λ2 then µR(xi, xk) = X3 where X3 ≥ Min [λ1, λ1]
For example, let X be set of cities. R be the fuzzy relation representing concept “very near”. Then
- R is reflexive since a city is certainly very near to itself ie, its membership value is 1 so it satisfies reflexivity.
- R is also symmetry since if ‘A’ is very near to ‘B’ by some degree 0.7 then city ‘B’ is certainly very near to A by the same degree.
- The relation R is non-transitivity because if city ‘A’ is very near to city ‘B’ by some degree 0.6, city ‘B’ is very near to city ‘C’ by 0.6 then it is not possible that city ‘A’ is very near to city ‘C’ by 0.6