# Notation for relations between sets

Are there any customary notation for: $(X\times Y) \cap f \ne \emptyset$ ($f$ is a binary relation, $X$ and $Y$ are sets)?

For example $\{ f(x) | x\in X \}$ is commonly denoted as $f[X]$. But are there any common notation for the above formula?

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From context, $X$ and $Y$ are sets. But what is $f$? – ncmathsadist Jul 15 '11 at 15:22
The notation $X \times Y \cap f$ is ambiguous. Do you mean $(X \times Y) \cap f$ or $X \times ( Y \cap f)$? A binary relation is a set of ordered pairs. The two formulas give different results. When you write formulas with different binary relations it is helpful to use parentheses to clarify the formula. – Jay Jul 15 '11 at 16:17
The clearest notation, in my opinion, would just be to write $\{(x,y) \in X \times Y : f(x,y)\}$. – Nate Eldredge Jul 15 '11 at 17:07

If $X$ is a subset of the domain of $f$ it is common to write $f|_X$ (sometimes $f\upharpoonright_X$). I have seen similar notations for restricting the range although these are not as common.
The most "concise" notation I can think of is $\displaystyle f|_{f^{-1}[Y]\cap X}$ (the subscript part is $f^{-1}[Y]\cap X$). That is to say, restrict $f$ to the preimage of $Y$ which is in $X$.
Regardless of the above, I think that the most concise way is to "Denote $g = (X\times Y)\cap f$ ..." which is clear and to the point. Many times unsophisticated notation yields the clearer results.