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.



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 $A\subseteq D$, $$R[A]=\left\{y\in D:\exists x\in D\big(\langle x,y\rangle\in R\big)\right\}\;.$$

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