Discussion
Consider a fuzzy set A defined on the interval X = [0, 10] of integers by the membership Junction
\(\mu_A(x) = \frac{x}{x+2}\)
Then the \(\alpha\) cut corresponding to \(\alpha\) = 0.5 will be
Answer: C
The alpha cut of a fuzzy set contains the elements whose membership is at least alpha. Here the membership is \(\mu_A(x) = x/(x+2)\) and alpha = 0.5, so the condition is
\[ \frac{x}{x+2} \ge 0.5 \]\[ x \ge 0.5x + 1 \Rightarrow x \ge 2 \]The values of x in the interval [0, 10] that satisfy this are 2, 3, ..., 10. For x = 1 the membership is 1/3, which is less than 0.5, and for x = 0 it is 0, so these are left out. The set is {2, 3, 4, 5, 6, 7, 8, 9, 10}, the third option. The empty set would hold only if no value reached 0.5.
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