Images Preserve Inclusions
Images Preserve Inclusions
For \(f : X \to Y\) and \(A_1, A_2 \subseteq X\)
\[ A_{1} \subseteq A_{2} \implies f(A_{1}) \subseteq f(A_{2})\]
Proof
Let \(y \in f(A_1)\). Therefore, there exists an \(x \in A_1\) such that \(f(x) = y\). Such an \(x\) is also in \(A_2\) by assumption. Therefore there exists an \(x \in A_2\) such that \(f(x) = y\), which means \(y \in f(A_2)\).