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Notation for image and preimage

Is there a standard notation for the function $F$?

$f:X\to Y$, define $F:2^X \to 2^Y$ as $F(S) = \{f(x)|x\in S\}$.

Recently I have to define this kind of function very often, and wonder there are already notations for this.

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    In some fields, one would write ${\rm map} f$ for $F$. It occurs to me that if you stated up front that you were going to write ${\mathcal P}(f)$ or ${\mathcal P}f$ for this, nobody would complain.2012-06-19
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    Since you’re using $2^X$ for the power set of $X$, you could establish a convention that if $f:X\to Y$, then $2^f:2^X\to 2^Y:S\mapsto f[S]$.2012-06-19
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    @Brian: That seems more likely to confuse than not. In general, given a function $f : X \to Y$, one gets a map of function spaces $Z^f : Z^Y \to Z^X$ going the other way!2012-06-19
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    I feel like in some category theory contexts, this is either referred to as $f^*$ or $f_*$, but I've forgotten which.2012-06-19
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    @Zhen Lin: If you say so: I’ve never seen the notation. (And on first exposure don’t much care for it, for whatever that may be worth.)2012-06-19
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    @Thomas: Upper stars indicate contravariance. Some people would write $f_*$ for this. I would write $f_!$, because $f_!$ is left adjoint to $f^* = f^{-1}$.2012-06-19
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    Maybe a modified version of Brian's suggestion: ${_2}f$...maybe chemists prefer this one.2012-06-19

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