Pre-Images Preserve Inclusions
Pre-Images Preserve Inclusions
For \(f : X \to Y\) and \(B_1, B_2 \subseteq Y\)
\[ B_1 \subseteq B_2 \implies f^{-1}(B_1) \subseteq f^{-1}(B_2)\]
Proof
\[\begin{align*}
x \in f^{-1}(B_1) &\iff f(x) \in B_1 \\
&\implies f(x) \in B_2 & \text{by assumption} \ B_1 \subseteq B_2 \\
&\iff x \in f^{-1}(B_2) \\
\end{align*}\]