There exists a notation to "equations" where the unknown to solve is a kind of relation?

Question

Sometimes when doing exercises I find myself trying to solve the next "equation" (I dont know a better name):

$$f(x)\operatorname{@} g(x),\quad\forall x\in A\tag{1}$$

where $\operatorname{@}$ is the unknown, and it represent a kind of relation, generally an order relation (sometimes an equivalence relation), and $f$ and $g$ are functions. A solution to $(1)$ exists when some relation, from a predefined set of relations, holds for all $x\in A$ (sometimes I found myself writing $\sim$ instead of $\operatorname{@}$ but this can be confused with the use of $\sim$ as an equivalence relation).

There is some more standard notation for these kind of "equations"? There is a formal term to express it instead of the informal term "equation" that I had used here?

Answer 1

To me, I would take it as a "predicate" regardless of how logicians would call it. What do we mean to solve the equation $3x = 5$ in the real numbers? We mean to find a real number $x$ such that $3x = 5$; in plain language, we mean to find a real number to which the predicate "$3$ times it equaling $5$" is applicable. In a similar spirit, to solve $f @ g$ on $A$ for $@$ in a given set $M$ means to find some $@ \in M$ such that $f@g$ on $A$; this means to find some point of $M$ to which the predicate "$f$ 'the point' $g$ on $A$" is applicable.

Answer 2

Probably no one will agree with me, but I think we should just stop using the word "equation" altogether. Lets just refer to $3x+1=4$ as a statement. We can also refer to it as a set so long as it's understood that implicitly, we're really referring to $\{x \in \mathbb{R} \mid 3x+1=4\}$. Instead of speaking of the "solutions" of $3x+1=4$, we can speak of the "elements" of $3x+1=4$. Also, note that the set $3x+1=4$ equals the set $x=1$, but the difference is that the latter statement is in normal form. So instead of saying "solve the equation", we can say "normalize the statement" or "put the statement in normal form." So the phrase you're looking for is probably: "Our goal is to normalize $f(x)\operatorname{@} g(x)$."