Consider a fuzzy set A defined on the interval X = [0, 10] of integers by the…

Concept

For a fuzzy set A on a universe X, the α-cut of A is the crisp (ordinary) set of all elements whose membership grade reaches at least the threshold α, that is A(α) = { x ∈ X : A(x) ≥ α }. Thresholding the membership function in this way converts a fuzzy set into an ordinary set. The strong α-cut instead uses the strict inequality A(x) > α, so unless a question says "strong", the ≥ form is intended.

Application

Here the universe X is the set of integers {0, 1, 2, …, 10}, the membership function is A(x) = x / (x + 2) and the threshold is α = 0.5, so the required α-cut is the set of integers x in X that satisfy x / (x + 2) ≥ 0.5.

1. Write the defining condition of the α-cut for this membership function: x / (x + 2) ≥ 0.5.

2. Every x in X satisfies x + 2 ≥ 2 > 0, so multiplying both sides by (x + 2) is safe and does not flip the inequality: x ≥ 0.5 (x + 2).

3. Expand the right-hand side: x ≥ 0.5x + 1.

4. Subtract 0.5x from both sides: 0.5x ≥ 1.

5. Multiply both sides by 2: x ≥ 2.

6. Keep only the integers of X that satisfy x ≥ 2, which gives the set {2, 3, 4, 5, 6, 7, 8, 9, 10}.

Cross-check

Writing A(x) = x / (x + 2) = 1 − 2 / (x + 2) shows the membership grade increases steadily with x, so once the threshold is met it stays met. Evaluating the grades near the boundary confirms where the crossover happens.

So the α-cut for α = 0.5 is {2, 3, 4, 5, 6, 7, 8, 9, 10}. The element x = 2 belongs to it because the α-cut is defined with ≥ and A(2) = 0.5 exactly; the strong 0.5-cut would drop it and give {3, 4, 5, 6, 7, 8, 9, 10}.