Fuzzy Logic . How to get the complement

Question

Hi, i came across the following fuzzy logic example about fuzzy logic.

Representing Age Problem 2-1. Fuzzy sets can be used to represent fuzzy concepts. Let U be a reasonable age interval of human beings.

U = {0, 1, 2, 3, ... , 100}

Solution 2-1. This interval can be interpreted with fuzzy sets by setting the universal space for age to range from 0 to 100.

Problem 2-2. Assume that the concept of "young" is represented by a fuzzy set Young, whose membership function is given by the following fuzzy set.

Young= FuzzyTrapeZoid [0 ,0 ,25 ,40]

All i want to understand is how i can get the Complement[Young]

Answer 1

The quick answer is that Complement[Young] = FuzzyTrapeZoid[25,40,100,100]. Here is an image to show (in red) Young, and the complement in green.

Were you looking for an algorithm to solve this?

edit: adding more:

A generic fuzzy trapezoid is: FuzzyTrapeZoid[A,B,C,D]

The membership value is 0 up to A, then ramps from 0 to 1 between A and B, stays at 1 from B to C, then ramps from 1 to 0 between C and D. see page 3 of this intro (warning! pdf)

Since the complement of a fuzzy set = 1 - the membership function, then you can pretty much see the values by inspection. For the original problem (which comes from Mathematica), the complement is a single function. For the generic one FuzzyTrapeZoid[A,B,C,D] you will need 2 trapeziods to make the complement: FuzzyTrapeZoid[0,0,A,B] + FuzzyTrapeZoid[C,D,100,100]

For the Young membership function, it is 1 up to 25, so the complement will be 0 up to 25 (this yields [25,x,x,x] where x is yet to be determined). Since the Young membership function ramps to 0 between 25 and 40, it is clear that the complement will ramp from 0 to 1 in the same range (this yields the [25,40,x,x] where x is yet to be determined). Finally, since the Young membership function is 0 from 40 to 100, the complement will be 1 in the same range, this gives [x,40,100,100] (we knew from before that x = 25).

If you were looking for some more formal proof, I'm sorry, I do proofs poorly as I come from the Captain Kirk school of math: I can see it, and I can jump to the right answer, but I can't tell you exactly how I did it.