Proving 2 sets are subsets of each other

**onemore** · April 19th 2010, 10:15 AM

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

**Failure** · April 19th 2010, 10:31 AM

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 $y\in f^{-1}(D_2)$ .

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