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Let $f:S\to T$. Let $S\subset X$ and $Y\subset T$. If $X=f^{-1}(Y)$, is it true that $f(X)\subset Y$? Is it true that $f(X)=Y$?

The first claim is true. Pick some $y_0\in f(X)$. Then there exists some $x_0\in X$ such that $f(x_0)=y_0$. By definition of inverse image, $X = \{s\in S: f(x)\in Y\}$. Thus, we have $f(x_0)\in Y$. That is, $y_0\in Y$. This implies that $f(X)\subset Y$.

But I'm not sure about the second claim?

3x89g2
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1 Answers1

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No, the second is true if $f$ is surjective but otherwise not. Take e.g. $f(x)=x^2$ and $Y=[-1,1]$.

H. H. Rugh
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