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Suppose I have a function $f:\mathbb{R}\rightarrow\mathbb{R}$ and a set of points $S\subseteq\mathbb{R}^2$. How can I concisely convey the following idea?

All points $(x,y)\in S$ such that $f(x)=y$

I was thinking of writing $S\cap f$, as if $f$ were also a set of points, but I was thinking this could be seen as abuse of notation (since $f$ is described as a function and not a set of points). Is there a more standard notation for this?

Ideally this notation should be applicable to non-real-valued functions. In reality, $f$ is actually a computer program mapping inputs to outputs, and $S$ is a set of desired input-output pairs, and I want to describe the subset of $S$ that is "consistent" with $f$'s behavior. However I'm using a function analogy to provide intuition first in my writing.

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  • $\begingroup$ This would be the intersection of the graph of $f$ with $S$. To my knowledge, there is no standard notation for it, although $\Gamma _f$ may be common. Then, you are looking at $\Gamma _f \cap S$. $\endgroup$
    – Suzet
    Jun 29, 2018 at 6:22
  • $\begingroup$ A function $f$ is a set of points, so $S\cap f$ should work. $\endgroup$ Jun 29, 2018 at 6:28
  • $\begingroup$ @Henrik Maybe in pure math, but note that $f$ is actually a placeholder for a computer program (i.e., the source code), which isn't usually considered a set of input-output pairs. This is why reviewers have said my notation was confusing/"abuse of notation". $\endgroup$
    – k_ssb
    Jun 29, 2018 at 6:33

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For $f :A \mapsto B$, I would set $G:=\{(x,f(x)),x \in A\}$ and then the set you want to define can be expressed as $G \cap S$.

Note that $G$ stands for graph.

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