Shorthand notation for "increases" and "decreases"

In fact, the issue @Angada is referring has nothing to do with proportionality and $\propto$ symbol.

It appears that there is no enough widely accepted mathematical notation for that, although some complex notations are developed for fuzzy logic and fuzzy reasoning that unfortunately collide wit logic symbols.

Anyway I think an easy convention for this is a necessity if we want to do operative reasoning based on qualitative rules. I found this lack of mathematical notation so irritating that a couple of years ago, I began to develop a convention that today use successfully for my own .

The model of expression I was run after was that of Bachmann-Landau O notation, so the only thing I would had to do is to choose a symbol "S" with the meaning: "S(x) is any arbitrary monotonic increasing function of x".

Finding an intuitive significant for a increasing function is not an easy issue, and perhaps this is the reason why this convention has (as long as I know) been never established. If not, try yourself to imagine a particular shape associated with increasing.

My first idea was to use a crescent moon as an icon for such functions, but unfortunately this wouldn't stand for people living in the southern hemisphere as moon appearance is reversed from there.

Then I abandoned the onomatopoeic approach and ended up with an hybrid of the symbol of Derivative ("D") and the positive sign ("+"), leading to "Đ" (crossed D ,D with stroke or dyet) that also reminds for Directly (but not necessarily proporcional) related.

The symbols Đ and its lowercase: đ are part of the croatian and vietnamese alphabets, and have unicode in latin U+0110 , and U+0111 respectively.

Now I use;

• $f(x)=Đ(x)$ or $f(x)\in \{Đ(x)\}$ for strictly monotonically increasing funcions,
• and $f(x)=đ(x)$ or $f(x)\in \{đ(x)\}$ for (relaxed) monotonically increasing functions.

That is:

• $f(x)=Đ(x) \equiv a>b \Longleftrightarrow f(a) > f(b)$
• $f(x)=đ(x) \equiv a>b \Longleftrightarrow f(a)\geq f(b)$

Monotonic decreasing funcions are then esaily denoted as $f(x)=-đ(x)$, or (strictly) $f(x)=-Đ(x)$

Some properties are:

• $Đ(Đ(x))=Đ(x)$
• $Đ(đ(x))=đ(x)$
• $f(x)=Đ(x)\Rightarrow f(x)=đ(x)$
• $Đ(-x)=-Đ(x)$
• $đ(-x)=-đ(x)$
• $đ(k x^n)=Sign(k) đ(x)$ , for any $n \ge 0$ $x=0$
• $đ(k x^n)=Sign(k) Sign(n) đ(x)$ , for any interval that does'nt contain $x=0$
• etc.

Of course, this could be combined with $\forall$ in order to restrict domains as for instance: $$f(x)=\left\{\begin{matrix} Đ(x) & \forall x \in (-\infty,2) \\ -đ(x)) & \forall x \in [2,4] \\Đ(x) & \forall x \in (4,\infty) \end{matrix}\right.$$

To respond to your question: in order to express: "As x increases, y decreases.", you could simply write $y=-Đ(x)$.

From the point of view of verbalization, $y=Đ(x)$ would be read as: "$y$ is directly related to $x$" (but not: "$y$ is directly proportional to $x$"); and $y=- Đ(x)$ as "$y$ is inversely related to $x$".

Of course this is only a proposal, and by now, it works well only for my own use. If you find it useful, I encourage you to use it. If so, I would appreciate a lot if You'd let me know.

Otherwise: if you know or find a simple notation more established, shorter, or more intuitive, please let me know and I will be very happy to adopt it.