Let's assume $A$ and $B$ are two discrete sets, and $A^{\prime}$ and $B^{\prime}$ are their subsets (i.e. $A^{\prime}\subseteq A$ and $B^{\prime}\subseteq B$). We have a mapping $f$ from $A$ to $B$ (i.e. $f:A\rightarrow B$).

If $B=f\left(A\right)$ and $B^{\prime}=f\left(A^{\prime}\right)$ hold, can we prove or disprove $B\setminus B^{\prime}=f\left(A\setminus A^{\prime}\right)$ ?