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copied!<p>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.<br> <img src="https://i.stack.imgur.com/Kht5V.jpg" alt="alt text"></p> <p>Were you looking for an algorithm to solve this?</p> <p><strong>edit</strong>: adding more: </p> <p>A generic fuzzy trapezoid is: <code>FuzzyTrapeZoid[A,B,C,D]</code> </p> <p><img src="https://i.stack.imgur.com/YuGuP.jpg" alt="alt text"></p> <p>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. <a href="http://www.cs.nott.ac.uk/~sxp/DSM/fuzzy_sets_intro.pdf" rel="nofollow noreferrer">see page 3 of this intro</a> (warning! pdf)</p> <p>Since the complement of a fuzzy set = <code>1 - the membership function</code>, then you can pretty much see the values by inspection. For the original problem (which comes from <a href="http://www.wolfram.com/products/applications/fuzzylogic/examples/age.html" rel="nofollow noreferrer">Mathematica</a>), the complement is a single function. For the generic one <code>FuzzyTrapeZoid[A,B,C,D]</code> you will need 2 trapeziods to make the complement: <code>FuzzyTrapeZoid[0,0,A,B] + FuzzyTrapeZoid[C,D,100,100]</code> </p> <p>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).</p> <p>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. </p>

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