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I have the following excercise:

$f : A\rightarrow B$ is a function. $\forall Y\subset B $ we have that $f(f(Y)⁻¹) \subset Y $. Could someone give me an example where the subset is true?.

Kind regards,

Phi.

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    Well, it's true for *any* example, so just pick some $f, A, B,$ and $Y$ and check it out. (Also you mean "$f^{-1}(Y)$," not "$f(Y)^{-1}$".) – 2017-02-13
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    I assume you meant $f(f^{-1}(Y))$ and I assume you meant **proper** subset. If so, take $A=\emptyset$ and any nonempty $B$. – 2017-02-13

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You mean a proper inclusion. Consider $A = B = \text{real numbers}$, $f(x) = x^{2}$, and $Y = B$.

We have that $f(f^{-1}(Y))$ only contains positive numbers, hence it is properly contained in $Y = B$.

More generally, any non-surjective $f$ will do, as $f(f^{-1}(B)) \subseteq f(A)$ will be properly contained in $B$.

As noted in a comment, you mean $f^{-1}(Y)$.