Operation on Fuzzy Sets

7/21/2019 Operation on Fuzzy Sets

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Let R and S denote two fuzzy relations in theCartesian space X x Y.

1. Union operation: The union of two fuzzyrelations R and S in the Cartesian space X x Y is

defined as μRY S(x, y) = max(μR(x, y),μs(x, y))2. Intersection operation: The intersection of two

fuzzy relations R and S in the Cartesian space X xY is defined as μRIS(x, y) =min(μR(x, y), μs(x, y))

3. Complement operation: The complementoperation of the fuzzy relation R is defined asμR(x, y) = 1 − μR(x, y)



Complement:A’(x) = 1- A(x) for every x in U

Intersection:(A I B)(x) = min[A(x),B(x)] for every x in U

Union:

(A Y B)(x) = max[A(x),B(x)] for every x in U



Complement



R: X x Y −→ [0,1] S: Y x Z −→ [0,1]

can be composed to one relation:

T : X x Z −→ [0,1] using the max and min operators for union and

intersection, can express the compositionoperation T = R o S by the corresponding

membership functions.



Observations:

Max-Min composition

R =0.1000 0.3000 0.5000 0.60000.3000 0.5000 0.8000 1.00000.2000 0.9000 0.7000 0.1000

S =

0 0.30000.9000 0.80000.4000 0.70000.2000 0.6000

T =

0.4000 0.60000.5000 0.70000.9000 0.8000



Given R: x × y and S: y1 × z as follows.

R =

0.1 0.3 0.5 0.60.3 0.5 0.8 10.2 0.9 0.7 0.1

S =0 0.30.9 0.80.4 0.70.2 0.6

Compute the max product composition T: x × yusing a MATLABprogram.



Given a fuzzy relation R=

Write a MATLAB program to compute max R^2and R^3.



A=[a1,a2], B=[b1,b2], C=[c1,c2], 0=[0,0], 1=[1,1]

Operation on Fuzzy Sets

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