Let $R \subseteq X \times Y$

Is there a commonly used term/notation for the functions $f:X\rightarrow \mathcal{P}(Y)$ and $g:Y\rightarrow \mathcal{P}(X)$ defined as follows?:

$$f(x) = \{ y \mid (x, y) \in R\}$$ $$g(y) = \{ x \mid (x, y) \in R\}$$

Since these values are a very elementary part of the structure of a relation, I'd think that this shows up commonly and might have a name.

If there is notation for $f$ then perhaps $g$ doesn't need different notation as $g$ is just $f$ on $R^{-1}$

  • $\begingroup$ The question is fine as it is but I think some more context about what prompted this question might help stir up some ideas $\endgroup$ – gen-ℤ ready to perish Jan 10 '19 at 19:52
  • $\begingroup$ I'm just using set theory to formalize something very niche (as it should be used) and I need to refer to these functions quite often so I defined my own terminology. Though I'd love to know if there was some standard terminology and use it. $\endgroup$ – Peeyush Kushwaha Jan 10 '19 at 19:54
  • $\begingroup$ In analogy to functions, I've seen $R(A) = \{y\in Y\mid (a,y)\in R\text{ for some }a\in A\}$ and $R^{-1}(B) = \{x\in X\mid (x,b)\in R\text{ for some }b\in B\}$. By abuse of notation, $R(\{x\})$ is often written $R(x)$, and context is meant to provide disambiguation where needed. I'm not sure about your universal quantification, given that you are putting it outside the set and out-of-place for logical notation. I'm guessing you mean that for each $x\in X$ you define $f(x)$ to be blah, etc. In any case, as this is not universal, I would suggest defining the notation briefly before using. $\endgroup$ – Arturo Magidin Jan 10 '19 at 20:06
  • $\begingroup$ @ArturoMagidin yes, that's what it was supposed to mean. I'll clarify. I like the notation. If you could put it up as an answer with some more information about where you've seen it being used, it would be a good answer. $\endgroup$ – Peeyush Kushwaha Jan 10 '19 at 20:08
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    $\begingroup$ I don't know about specific notation, but $f(x)$ and $g(y)$ are often called, respectively, the $x$-vertical section of $R$ and the $y$-horizontal section of $R.$ See this google search and this other google search. $\endgroup$ – Dave L. Renfro Jan 20 '19 at 9:55

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