Let \(A_{\alpha_0}\) denotes the \(𝛼\)-cut of a fuzzy set \(𝐴\) at…
2019
Let \(A_{\alpha_0}\) denotes the \(𝛼\)-cut of a fuzzy set \(𝐴\) at \(\alpha_0\). If \(\alpha_1 < \alpha_2\), then
- A.
\(A_{\alpha_1} \supseteq A_{\alpha_2}\) - B.
\(A_{\alpha_1} \supset A_{\alpha_2}\) - C.
\(A_{\alpha_1} \subseteq A_{\alpha_2}\) - D.
\(A_{\alpha_1} \subset A_{\alpha_2}\)
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Correct answer: A
Reason: By definition the α-cut of a fuzzy set A is A_α = { x | μ_A(x) ≥ α }.
Take any x ∈ A_{α2}. Then μ_A(x) ≥ α2.
Since α1 < α2, the inequality μ_A(x) ≥ α2 implies μ_A(x) ≥ α1, so x ∈ A_{α1}.
Therefore every element of A_{α2} is in A_{α1}, i.e. A_{α2} ⊆ A_{α1}, which is equivalent to A_{α1} ⊇ A_{α2}.
Note: The inclusion may be non-strict (equality can occur) if there are no elements with membership values between α1 and α2.
Conclusion: If α1 < α2, then A_{α1} ⊇ A_{α2}.
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