Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is there a name for constructing a set from a relation (or, more generally speaking, from a set of pairs that are tuples)? For example, let $R = \{(0, 1), (1, 2), (2, 3)\}$; if you collect all the second elements in the tuples (pairs) you get $A = \{1, 2, 3\}$.

Is there a name for the subset of a relation for which the first element in each list is a certain $x$? This means that there is some sort of filter operation $f(x, R)$ that gives you $R'$, where $R'$ is the subset of $R$ with all the pairs in $R'$ having $x$ as its first element.


share|improve this question
add comment

1 Answer

The set $A$ in your first example is the range of $R$, commonly written $\operatorname{ran}R$ or $\operatorname{Rng}R$. I’m not aware of a standard notation for what you want in your second example. If $R$ is a relation on $D$, that set is $R\cap\big(\{x\}\times D\big)$. If I were going to need it at all often in something, I’d define some reasonably convenient ad hoc shorthand notation for it, probably $R_x$.

If you simply want the second components of all of the pairs in $R$ having $x$ as first component, that can be written $R[\{x\}]$. More generally, if $A\subseteq D$, $$R[A]=\left\{y\in D:\exists x\in D\big(\langle x,y\rangle\in R\big)\right\}\;.$$

share|improve this answer
add comment

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.