For a trapezoidal membership function with parameters a=0, b=2, c=6, and d=8, if b is changed to 4, which of the following statements are true: At x=1 and x=7, the degree of membership will be 0.5 At x=2 and x=7, the degree of membership will be 0.5 At x=4 and x=7, the degree of membership will be 0.5 None of the above
Question
For a trapezoidal membership function with parameters a=0, b=2, c=6, and d=8, if b is changed to 4, which of the following statements are true: At x=1 and x=7, the degree of membership will be 0.5 At x=2 and x=7, the degree of membership will be 0.5 At x=4 and x=7, the degree of membership will be 0.5 None of the above
Solution
The trapezoidal membership function is defined as follows:
f(x;a,b,c,d) = 0, if x <= a or x >= d = (x - a) / (b - a), if a <= x <= b = 1, if b <= x <= c = (d - x) / (d - c), if c <= x <= d
Given the parameters a=0, b=2, c=6, and d=8, if b is changed to 4, the function becomes:
f(x;0,4,6,8) = 0, if x <= 0 or x >= 8 = (x - 0) / (4 - 0), if 0 <= x <= 4 = 1, if 4 <= x <= 6 = (8 - x) / (8 - 6), if 6 <= x <= 8
Now, let's evaluate the function at the given x values:
At x=1, f(1;0,4,6,8) = (1 - 0) / (4 - 0) = 0.25, not 0.5.
At x=7, f(7;0,4,6,8) = (8 - 7) / (8 - 6) = 0.5, which is correct.
At x=2, f(2;0,4,6,8) = (2 - 0) / (4 - 0) = 0.5, which is correct.
At x=4, f(4;0,4,6,8) = 1, not 0.5.
So, the statement "At x=2 and x=7, the degree of membership will be 0.5" is true.
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trapezoidal rule
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