Let $\text{S}$ and $\text{T}$ be two non-empty sets and $f : \text{S} \rightarrow \text{T}$ be a function such that $f (\text{A} \cap \text{B}) = f (\text{A}) \cap f(\text{B})$ for all subsets $\text{A}$ and $\text{B}$ of $\text{S}$. Then
-
there exist $\text{A} \subset \text{S}$ such that $f^{-1} f(\text{A}) \neq \text{A}$
-
$f$ is one- to- one
-
there exist disjoint subsets $\text{A}, \text{B}$ of $\text{S}$ such that $f(\text{A}) \cap f(\text{B}) \neq \phi $
-
none of the above statements is necessarily true.