I am a little bit confused about fuzzy set theory extension principle description. Most sources define extension principle in a following way:
Suppose $f:X\to Y$ and fuzzy set $A\in X$ such that $A=\sum_{x\in X}\frac{\mu_{A}(x)}{x}=\frac{\mu_{A}(x_{1})}{x_{1}}+...+\frac{\mu_{A}(x_{n})}{x_{n}}$.
Then extension principle states that the image of A under the mapping f can be expressed as a fuzy set $B=f(A)=\frac{\mu_{A}(f(x_{1}))}{x_{1}}+...+\frac{\mu_{A}(f(x_{n}))}{x_{n}}$
My question is how could fuzzy set be defined as a sum of rational numbers devided by vector? So in my consciousness it contradicts definition of fuzzy sets as a set of ordered pairs $(x,\mu_{A}(x)), x\in X$.
Will be very gratefull for explaining this problem or providing another, more easy to understand definition of extension principle!
Klir & Folger [1] says that $B=f(A)=\frac{μ(x_1)}{f(x_1)}+...+\frac{μ(x_n)}{f(x_n)}$, that is, the extension principle applies maps that map a crisp set to another, to fuzzy sets, turning a fuzzy subset $A$ to a fuzzy subset $B$.
If more than one element of $X$ is mapped by $f$ to the same element $y \in Y$, then the maximum of the grades is chosen as grade for $y$ in $f(A)$.
[1] Klir & Folger, Fuzzy Sets, Uncertainty and Information, 1988.