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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.

Johannes Titz
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    Please could you give the context in which you found the expression "monotonic relation"? It seems to be used in an unformal way in some statistics papers, typically "a monotonic relation between two variables" meaning than when one variable grows the other grows, but without a precise mathematical definition. In particular, as it is a statistical dependency, it is not systematically true, i.e. it is possible to find pairs $(a,b), (c,d)$ where $a < b$ and $c > d$. – Jean-Armand Moroni Oct 27 '23 at 12:48
  • Thanks for adding the context. Now, as you wrote, you are free to come with any definition of isotonic relation you want, and say "this is my definition of isotonic". The only point to ensure is that, when the relation is a function, the definition must be equivalent to the usual definition for functions. Which still leaves plenty of possibilities. – Jean-Armand Moroni Oct 30 '23 at 10:17
  • Ok, but would you say you can follow the formal notation? Is it intelligible? – Johannes Titz Oct 31 '23 at 09:44
  • Which notation? I don't understand what you call "formal notation" in the question text, sorry for that. – Jean-Armand Moroni Oct 31 '23 at 10:19
  • Well, I am just asking if the expressions that are used make sense or if a different way of writing it would be better. For instance, a way that is more accurate or more common to use. My fear is that other people might not understand it or find it peculiar. – Johannes Titz Oct 31 '23 at 10:36
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    If you are talking about representing a relation as a set of pairs, i.e. $R = { (0,0),(0,1),(1,1) }$, and the monotonicity criterion, i.e. $\forall (i_1, i_2), (i_1', i_2') \in R, \quad i_1 < i_1' \implies i_2 \leq i_2'$, that perfectly makes sense and any mathematician would understand it. – Jean-Armand Moroni Oct 31 '23 at 10:43
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    I think you could (equivalently) say that a relation $R$ is isotonic $\iff$ every function $F \subseteq R$ is isotonic. – mjqxxxx Nov 02 '23 at 16:43

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