hello everyone I have question
Q Let $f:A\rightarrow B$ and $C\subset B,D\subset B$. Prove that $f^{-1}(C\backslash D)=f^{-1}(C)\backslash f^{-1}(D)$?
Notice,
(the inverse image not inverse function)
I hope some one can prove thank you
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Ben West
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leena adam
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1Here $f^{-1}$ doesn't denote the inverse of $f$ (which may not even exist), but rather $f^{-1}(Y)={a\in A: f(a)\in Y}$, for any $Y\subseteq B$. – Git Gud Apr 25 '13 at 21:26
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See also: Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$ – Martin Sleziak Sep 13 '19 at 12:43
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$x \in f^{-1}(C\backslash D) \Leftrightarrow f(x) \in C\backslash D \Leftrightarrow f(x)$ is in $C$ and not in $D \Leftrightarrow x \in f^{-1}(C)$ and $ x \notin f^{-1}(D) \Leftrightarrow x\in f^{-1}(C)\backslash f^{-1}(D)$
justt
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To prove equality of two sets show that each is contained in the other. So for one direction assume $x \in f^{-1}(C \setminus D)$. This means $f(x) \in C \setminus D$, so $f(x) \in C$ but $f(x) \notin D$. Thus $x \in f^{-1}(C)$ but $x \notin f^{-1}(D)$, giving $x \in f^{-1}(C) \setminus f^{-1}(D)$.
That proves $f^{-1}(C \setminus D) \subseteq f^{-1}(C) \setminus f^{-1}(D)$. Now I leave it to you to emulate what I've done and prove the other direction. Once you have both directions you will have shown that $f^{-1}(C \setminus D) = f^{-1}(C) \setminus f^{-1}(D)$.
Jim
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