Prove or disprove:
Let $f : X \to Y$ be an onto function and let $A$ and $B$ be subsets of $Y$. If $f^{−1}(A) \subseteq f^{−1}(B)$, then $A \subseteq B$.
Is this statement true? I try to come with counterexample but I cannot; I think it is true but I am not sure.
Could you please confirm that for me ?
The claim is true. Let $a \in A$. Since $f$ is onto, there exists $x \in X$ such that $f(x)=a$. But that means that $x \in f^{-1}(A)$ and thus $x \in f^{-1}(B)$. Therefore, $a=f(x) \in B,$ as desired.
The claim is not true if $f$ is not onto. To see that, let $X=Y=\mathbb{R}$ and let $A=[-2,2],$ $B=[-1,1],$ and $f(x)=0$ for all $x.$ Then clearly $X=f^{-1}(A)=f^{-1}(B)$ but $A$ is not a subset of $B$.