# Notation for using a relation as a function from an element to set?

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 $$R^{-1}$$

• The question is fine as it is but I think some more context about what prompted this question might help stir up some ideas – gen-ℤ ready to perish Jan 10 '19 at 19:52
• 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. – Peeyush Kushwaha Jan 10 '19 at 19:54
• 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. – Arturo Magidin Jan 10 '19 at 20:06
• @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. – Peeyush Kushwaha Jan 10 '19 at 20:08
• 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. – Dave L. Renfro Jan 20 '19 at 9:55