Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.