Example where an inverse function does not equal the elements

Question

I'm trying to find an example of a function $f: A \to B$ and $X \subset A$ so that $f^{-1}(f(X)) \ne X$, and similarly where $Y \subset B$ so that $f(f^{-1}(Y)) \ne Y$.

I thought to have $f = x^2$, which has no inverse, thus making it vacuously true that $f^{-1}(f(X)) \ne X$ and $f(f^{-1}(Y)) \ne Y$, but that seems like a copout. Any help is greatly appreciated.

Answer 1

Consider $f : \mathbb{R} \rightarrow \mathbb{R}$ defined for all $x \in \mathbb{R}$ by $f(x)=0$, and $X = \lbrace 0 \rbrace$ and $Y = \mathbb{R}$.

Answer 2

Here $f^{-1}$ does not denote inverse function of $f$ but inverse image of a set, which is defined even when $f$ is not invertible. In fact your search for an invertible example will be in vain as for invertible $f,f(f^{-1}(Y))=Y$ and $f^{-1}(f(X))=X$ for all $X\subseteq A,Y\subseteq B$.

$y(x):\Bbb R\to\Bbb R,y=x^2$ is a valid example. Take $X=\{1\}\implies f^{-1}(f(X))=f^{-1}(\{1\})=\{\pm1\}$.

Take $Y=\{0,-1\}$. Then $f(f^{-1}(Y))=f(\{0\})=\{0\}$.