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Math Help - Set Theory Function Proof

  1. #1
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    Set Theory Function Proof

    Hi guys,

    Can you please check this basic set theory/function proof?

    \textbf{Theorem. }f^{-1}((\overline{f(S)}) \subseteq \overline{S}

    \emph{Proof. } Let a \in f^{-1}(\overline{f(S)}). Then f(a) = b for some b \in \overline{f(S)}. So we have that  b \notin f(S). Now b \notin f(S) implies that, for every k \in A such that f(k) = b, k \notin S, in particular, x. Thus x \notin S, hence x \in \overline{S} so that f^{-1}(\overline{f(S)}) \subseteq \overline{S}
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  2. #2
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    Quote Originally Posted by james121515 View Post
    Can you please check this basic set theory/function proof?
    \textbf{Theorem. }f^{-1}((\overline{f(S)}) \subseteq \overline{S}
    \emph{Proof. } Let a \in f^{-1}(\overline{f(S)}). Then f(a) = b for some b \in \overline{f(S)}. So we have that  b \notin f(S). Now b \notin f(S) implies that, for every k \in A such that f(k) = b, k \notin S, in particular, x. Thus x \notin S, hence x \in \overline{S} so that f^{-1}(\overline{f(S)}) \subseteq \overline{S}
    I think it can be shorter.
    a \in f^{-1}(\overline{f(S)}) implies that f(a)\notin f(S)
    So that a\notin S or a\in \overline{S}.
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