True or false: For all subsets $ A $ and $ B $ of $ X $, if $ A \subseteq B $, then $ f[A] \subseteq f[B] $. I am trying to determine if the following is true or false: > For all subsets $ A $ and $ B $ of $ X $, if $ A \subseteq B $, then $ f[A] \subseteq f[B] $. My guess is this would be true, because if $ A \subseteq B $, then $ f[A] \subseteq f[B] $. Can anybody corroborate my understanding, or tell me where I may have gone wrong? All help greatly appreciated!

Assume $A \subseteq B$. Let $y \in f(A)$. Then $\exists x \in A : f(x)=y$. But $A \subseteq B$, so if $x \in A$, then $x \in B$. Therefore, $\exists x \in B : f(x) = y$ so that $y \in f(B)$ as required. We conclude that $f(A) \subseteq f(B)$.