This is my first question on math stackexchange, so please be gentle.
I have different relations and need to check if they are monotonic. For instance:
$$R = \{ (0,0),(0,1),(1,1) \} $$
Is it formally correct to write the monotonicity criterion as:
$$\forall (i_1, i_2), (i_1', i_2') \in R \quad i_1 < i_1' \implies i_2 \leq i_2'$$
Note that the relation is not a function.
I searched the web and stackexchange, but was not able to find this particular case. Usually, monotonicity is discussed in the context of functions. Maybe I also missed it because I am not using the correct terminology. I have no math background, only programming. Sorry.
Edit: Adding more context as per Jean-Armand's request:
The starting point for me was monotonic functions, which can be easily defined via differential slopes. E.g. $$y=f(x)$$ is monotonically increasing when $$\frac{dy}{dx} \geq 0$$
The problem is that there are cases where y is not a function. So I need a more general notation. The idea is the same: when one variable grows, the other never decreases. Indeed, to be more specific, I just need the isotonic case, the antitonic case does not count for me.
I am not entirely sure if monotonicity can (or should) be defined that way for relations, but I want to show that $R$ is non-decreasing. And, for instance, $R_2 = \{(0,1),(1,0)\}$ is decreasing (not isotonic). Of course I could just say "this is my definition of isotonic", but does the notation make sense formally? Or would it be hard to read or just perceived as unusual?
I do not care about statistics. Either the relation is isotonic or not. Measurement error would be treated separately.