Math Help - Proving 2 sets are subsets of each other

1. Proving 2 sets are subsets of each other

Prove: If $f: A \rightarrow B$ and $D_1 \subset D_2$ are subsets of B, then $f^{-1} (D_1) \subset f^{-1} (D_2)$.

2. Originally Posted by onemore
Prove: If $f: A \rightarrow B$ and $D_1 \subset D_2$ are subsets of B, then $f^{-1} (D_1) \subset f^{-1} (D_2)$.
It suffices to show that any $y\in f^{-1}(D_1)$ happens to be an element of $f^{-1}(D_2)$ as well:
So let's assume that $y\in f^{-1}(D_1)$. This means that $f(y)\in D_1$. Because of $D_1\subseteq D_2$, it follows that $f(y)\in D_2$. But this is equivalent to saying that $y\in f^{-1}(D_2)$.