"Preimage" of a binary relation

Question

Consider the binary relation $R \subseteq X \times Y$. Is there a standard name and notation for the set $X' = \{x\ |\ (x, y) \in R\}$?

• ProofWiki calls $X'$ the preimage of $R$, denoted as $\operatorname{Im}^{-1}(R)$.
• This site calls $R$ a correspondence, calls $X$ the predomain of $R$ and calls $X'$ the domain of $R$.

Both conventions seem to make sense, although $X$ is generally called the domain of $R$, which I prefer as well. But the notation $\operatorname{Im}^{-1}(R)$ is rather clunky and unwieldy - not to mention that I hardly see it elsewhere - and I would prefer to use something simpler like $\operatorname{dom} R$ or $\operatorname{pre} R$, if these are accepted.

Of course, I should follow the standard notation, but I can't seem to find a consensus online!

Answer 1

I claimed in a comment that $\operatorname{dom}(R)$ is standard notation. I was asked by the OP to give out some references to support this claim. I present them below.

From D.J. Velleman's How to Prove It: A Structured Approach:

From Elements of Set Theory by H. B. Enderton:

And finally, from Thomas Jech's classic Set Theory:

Answer 2

In:

J.Riguet, "Relations binaires, fermetures, correspondances de Galois." Bulletin de la Société Mathématique de France, 76 (1948), pp.114-155

$X'$ is denoted as ${\rm pr_1}R$ (the first projection).