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Math Help - Proving 2 sets are subsets of each other

  1. #1
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    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).
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    Quote Originally Posted by onemore View Post
    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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