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!