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I'm learning fuzzy logic and i don't find many examples explaining Zadeh's extension principle I found this one but i don't know how to solve it. Can you help me ?

Let us consider two fuzzy subsets $A$ and $B$ defined by their membership functions $\mu_{A},\mu_{B}:\{1,2,3,4,5\}\rightarrow\{0, α, β, 1\}$, where $0 < \alpha < \beta < 1$ and

$A = \sum_{i=1}^{5}\mu_{A}(x_{i})/x_{i}=0/1 + \alpha/2 + 1 / 3 + 1 / 4 + α / 5$

$B = \sum_{i=1}^{5}\mu_{B}(x_{i})/x_{i}=0/ 1 + 0 /2 + \beta /3 + 1 / 4 + \beta/ 5$

(This is the standard notation to represent a fuzzy set with countable support).

Using the principle extension of Zadeh : suppose $f$ is a function with $n$ arguments that maps a point in $\omega$ to point in $V$, determine the membership function of $A + B$ and $\min\{A,B\}.$

Thanks.

Gustavo Alves
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Amine
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  • It's been sometime, and I just uploaded some corrections to your question, as to make it more comprehensible. What do you mean by a function $f:\omega\rightarrow V$? What are these two sets? – Gustavo Alves Apr 10 '24 at 15:08

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