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If A is the fizzy set defined by \[A=\frac{0.5}{x_{1}}+\frac{0.4}{x_{0}}+\frac{0.7}{x_{3}}+\frac{0.8}{x_{4}}+\frac{1}{x_{5}} \] List all α cuts of A.
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Solution:

$\alpha - level \hspace {0.1cm}sets:$ It contains the values whose membership values is greater than or equal to $\alpha$

$A_{\alpha=0.4}=\{ x_1, x_0,x_3,x_4,x_5\}\\ A_{\alpha=0.5}=\{ x_1,x_3,x_4,x_5\}\\ A_{\alpha=0.7}=\{ x_3,x_4,x_5\}\\ A_{\alpha=0.8}=\{x_4,x_5\}\\ A_{\alpha=1}=\{x_5\}\\ $

Strong $\alpha-level \hspace{0.1cm}sets:$ It contains the values whose membership values is greater than $\alpha$

$A_{\alpha=0.4}={ x_1,x_3,x_4,x_5} A_{\alpha=0.5}={ x_3,x_4,x_5}\\ A_{\alpha=0.7}=\{ x_4,x_5\}\\ A_{\alpha=0.8}=\{x_5\}\\ A_{\alpha=1}=\{\}$

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