Quidditch wrote: Let $f$ be a function on a set $X$. Prove that $$f(X-f(X))=f(X)-f(f(X)),$$where for a set $S$, the notation $f(S)$ means $\{f(a) | a \in S\}$. Wrong. Choose as counter-example $X=\{1,2\}$ and $f(x)=1\quad\forall x\in X$

How is subtraction defined?

Maybe $X-f(X)$ means $X \symbol{92} f(X)$ the elements $b$ in $X$ but not in $f(X)$ The functional equation is $f(X \symbol{92} f(X)) = f(X) \symbol {92} f(f(X))$ for $X$ in P(X) where P(X) the set of all subset of X

pco's still remain a counterexample; $f(\{1,2\} \backslash \{1\})=f(\{2\})=\{1\}$ while RHS is $\{1\} \backslash \{1\}$