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Fuzzy Logic
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Example 3: Finding the Disjunctive Sum

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

RMat = {{.8, .3, .5, .2}, {.4, 0, .7, .3}, {.6, .2, .8, .6}}
;

Smat = {{.9, .5, .8, 1}, {.4, .6, .7, .5}, {.7, .8, .8, .7}}
;

R = FromMembershipMatrix[RMat] ;

S = FromMembershipMatrix[Smat] ;

FuzzyPlot3D[R, S, ShowDots -> True] ;

[Graphics:HTMLFiles/sum_6.gif]

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.

DisSum = Union[Intersection[R, Complement[S]],
Intersection[Complement[R], S]] ;

FuzzyPlot3D[DisSum] ;

[Graphics:HTMLFiles/sum_9.gif]

ToMembershipMatrix[DisSum] // MatrixForm

( 0.19999999999999996`   0.5`
0.5`                   0.8`                 )    0.4`                   0.6`                   0.30000000000000004`
0.5`    0.4`                   0.8`                   0.19999999999999996`   0.4`

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