Consider a fuzzy set old as defined below Old = {(20, 0.1), (30, 0.2), (40,…
2014
Consider a fuzzy set old as defined below
Old = {(20, 0.1), (30, 0.2), (40, 0.4), (50, 0.6), (60, 0.8), (70, 1), (80, 1)}
Then the alpha-cut for alpha = 0.4 for the set old will be
- A.
{(40, 0.4)}
- B.
{50, 60, 70, 80}
- C.
{(20, 0.1), (30, 0.2)}
- D.
{(20, 0), (30, 0), (40, 1), (50, 1), (60, 1), (70, 1), (80, 1)}
Attempted by 71 students.
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Correct answer: D
Definition: The alpha-cut for a fuzzy set at level alpha is the set of all elements whose membership degree is greater than or equal to alpha (i.e., mu(x) ≥ alpha).
20 has membership 0.1, which is less than 0.4, so exclude 20.
30 has membership 0.2, which is less than 0.4, so exclude 30.
40 has membership 0.4, which is equal to 0.4, so include 40.
50 has membership 0.6 ≥ 0.4, include 50.
60 has membership 0.8 ≥ 0.4, include 60.
70 has membership 1 ≥ 0.4, include 70.
80 has membership 1 ≥ 0.4, include 80.
Conclusion: The alpha-cut for alpha = 0.4 is the crisp set {40, 50, 60, 70, 80}.
Note: None of the provided options exactly matches this correct alpha-cut. One option omitted 40, and another incorrectly displayed converted membership values (0 or 1) instead of listing the elements that meet the threshold.