Proof of a property of fuzzy extension principle

Question

I'm looking for a proof of the following property of the fuzzy extension principle:

$$ B \supseteq f(f^{-1}(B)) $$

($f: X\rightarrow Y$ is an arbitrary crisp function, and $B$ is a fuzzy set). I think if f is a function, the two sides would be equal. But I see this property on page 47 of the book by Klir and Yuan.

Thank you.

Answer 1

"I think if $f$ is a function, the two sides would be equal"

Not really, look at the function $f:\mathbb R\rightarrow \mathbb R$ defined by $f(x)=x^2$ and $B=[-1,0)$. Since $f^{-1}(B)=\emptyset$, it follows that $f(f^{-1}(B))=\emptyset$.

Answer 2

$f\left(f^{-1}\left(B\right)\right)\subseteq B$ is always true, which is not the case for $B\subseteq f\left(f^{-1}\left(B\right)\right)$ (see the answer of 5xgum). It is true for each $B$ if and only if $f$ is surjective.