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    $A \subseteq f^{-1}(f(A))$, with equality iff $f$ is injective

    P-F12IN

    Area
    Prelim
    Topics
    Functions and Relations
    Seen at
    UGA
    Years
    2012
    Status
    draft

    Let \(f: X \to Y\) be a (not necessarily invertible) function, and \(A \subseteq X\).

    1. Prove that \(A \subseteq f^{-1}(f(A))\).

    2. Prove that if \(f\) is injective (one-to-one) then \(A = f^{-1}(f(A))\).

    3. Give an example for which \(A \neq f^{-1}(f(A))\).

    Solution

    1. If \(x\in A\), then \(f(x)\in f(A)\), so by definition \(x\in f^{-1}(f(A))\). Thus \[ A\subseteq f^{-1}(f(A)). \]

    2. Suppose \(f\) is injective and \(x\in f^{-1}(f(A))\). Then \(f(x)\in f(A)\), so there exists \(a\in A\) with \(f(x)=f(a)\). Injectivity gives \(x=a\in A\). Thus the reverse inclusion holds and \[ A=f^{-1}(f(A)). \]

    3. Let \(X=\{0,1\}\), \(Y=\{0\}\), let \(f\) be constant, and take \(A=\{0\}\). Then \[ f^{-1}(f(A))=X\ne A. \]

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