Example 3: Finding the Disjunctive Sum

Problem. Find the disjunctive sum of the two fuzzy relations defined here.

The disjunctive sum of fuzzy relations R and S in the universal space VxW can be found with the following formula.

DisSum = (R ∩ S') ∪ (R' ∩ S)

The disjunctive sum is thus a relation in VxW that has the following property:

For all (v, w) in VxW, DisSum(v, w) =
    Max(Min(R(v, w), 1 - S(v, w)), Min(1 - R(v, w), S(v, w)))

Solution. We can find disjunctive sum of the two fuzzy relations R and S by using the formula derived earlier and some of the functions from Fuzzy Logic.