Hey,

Given that $\displaystyle f: X \rightarrow Y$ and that $\displaystyle A\subset X$ and $\displaystyle B \subset Y$ then I have to show that:

$\displaystyle f(f^{-1}(B)) \varsubsetneq B $.

I have done the following:

Assume that $\displaystyle y \in f(f^{-1}(B)) \Leftrightarrow$

$\displaystyle x \in f^{-1}(B)$ such that $\displaystyle f(x)=y \Leftrightarrow$

$\displaystyle y=f(x) \in B \Leftrightarrow$

$\displaystyle f(f^{-1}(B)) \subset B$

What remains is to show that $\displaystyle f(f^{-1}(B)) \neq B $. Would a counterexample be sufficient?

Thanks.