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Given some map $f : A \to B$, the distinction between the use of $\to$ and $\mapsto$ is usually clear: the former is to indicate the map is from $A$, the domain, to $B$, the range (or codomain), while the latter is to express the image of an element, say $a \mapsto b$ for $a \in A$ and $b \in B$.

However, is there a common usage of one or the other (or neither) when describing the image of a specific subset of $A$?

For example, let $A = B = \mathbb{Z}$ and $f : A \to B$ given by $f(x) = x^{2}$, with $S = \{1, 2, 3\}$ a subset of $A$. Then we can write $f(S) = \{1, 4, 9\}$ to express the image of $S$ under $f$, but would it be correct to write this with arrows as $$ S \to \{1, 4, 9\} \hspace{20pt}\text{or}\hspace{20pt} S \mapsto \{1, 4, 9\}? $$ My guess would be the former because some might read the latter as "the element $S$ maps to the set of three elements $\{1, 4, 9\}$", but I would like some clarification.

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You would write $f(S) = \{1,4,9\}$.

There is no reason for one of these other notations, since they are not more succinct. Furthermore they are unclear.

Using $\mapsto$ would seem to imply that $S$ is an element of the domain and gets sent to $\{1,4,9\}$, which is not technically what you are wanting (you could be looking at the function induced by $f$ on the power set of $A$, but that is an awfully complicated bit of machinery for something so elementary).

Using $\to$ is also awkward; $S \to \{1,4,9\}$ lacks a bit of context, since the function $f$ is not appearing in this statement. Writing $f: S \to \{1,4,9\}$ would come across as if you are redefining your symbol $f$ to refer to a new function with domain $S$.

TLDR; don't use $\to$ or $\mapsto$ to denote the image of a set under $f$.

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It's still a function that maps one subset onto another; $S$ isn't an element, so I wouldn't use $\mapsto$. So $S \to \{...\}$ is fine. I think what you mean is clear. Another option is, referencing the map $f$, to write $f|_S$ to denote $f$ restricted to the subset $S$ of the domain of $f$.

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None of $$S \to \{1, 4, 9\} \hspace{20pt}\text{or}\hspace{20pt} S \mapsto \{1, 4, 9\}$$ is used because both are ambiguous.

The common notation is $f[S]= \{1,4,9\}$

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It's a matter of convention, but I've never seen it. The only notation that I am aware of is the one that you mentioned: $f(S)$.

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