Fuzzy Relations
Introduction:-Fuzzy Relations describe the degree of association of the elements; Example : “x is approximately equal to y”. Fuzzy relations offer the capability to capture the uncertainty and vagueness in relations between sets and elements of a set. Fuzzy Relations make the description of a concept possible. Fuzzy Relations were introduced to supersede classical crisp relations; It describes the total presence or absence of association of elements.
Fuzzy relation is a generalization of the definition of fuzzy set from 2-D space to 3-D space.
Fuzzy relation :-Consider a Cartesian product
A x B = { (x , y) | x ∈ A, y ∈ B }
Where A and B are subsets of universal sets U1 and U2.
Fuzzy relation on A x B is denoted by R or R(x , y) is defined as the set
R = { ((x , y) , µR (x , y)) | (x , y) ∈ A x B , µR (x , y) ∈ [0,1] }
Where µR (x , y) is a function in two variables called membership function.
− It gives the degree of membership of the ordered pair (x , y) in R associating with each pair (x , y) in A x B a real number in the interval [0 , 1].
− The degree of membership indicates the degree to which x is in relation to y.
Forming Fuzzy Relations :-Assume that V and W are two collections of objects.
A fuzzy relation is characterized in the same way as it is in a fuzzy set.
− The first item is a list containing element and membership grade pairs,
{{v1, w1}, R11}, {{ v1, w2}, R12}, ... , {{ vn, wm}, Rnm}}.
Where { v1, w1}, { v1, w2}, ... , { vn, wm} are the elements of the relation are defined as ordered pairs, and { R11 , R12 , ... , Rnm} are the membership grades of the elements of the relation that range from 0 to 1, inclusive.
− The second item is the universal space; for relations, the universal space consists of a pair of ordered pairs,
{{ Vmin, Vmax, C1}, { Wmin, Wmax, C2}}.
Where the first pair defines the universal space for the first set and the second pair defines the universal space for the second set.