Tuesday, October 4, 2016

functions - Show $f^{-1}(A^c)=(f^{-1}(A))^c$


Let $f: X \to Y$, and $A\subseteq Y$. Show that $f^{-1}(A^c)=(f^{-1}(A))^c$


I know how to prove that $f^{-1}(A^c)\subseteq(f^{-1}(A))^c$, but stuck on proving $(f^{-1}(A))^c\subseteq f^{-1}(A^c)$. Could someone help with this step please? Thanks.


Answer



Suppose $x\in (f^{-1}(A))^c$. Then $x\in X$ and $x\notin f^{-1}(A)$. Thus $f(x)\notin A$, and of course $f(x)\in Y$, so $f(x)\in A^c$. Therefore $x\in f^{-1}(A^c)$.


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