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I want to write out something like:

"As $x$ increases, $y$ decreases."

Is there a standard symbolic notation for this, such as an up arrow and a down arrow? (And if you can tell me how to write it in latex, that would be awesome, too).

Thanks!

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1  
If $y$ is a function of $x$, you could just call it "strictly decreasing." This would probably be preferable to using symbols. – anon Feb 23 '12 at 7:06
up vote 1 down vote accepted

Inverse proportionality means that $y=\frac{k}{x}$ for some constant $k$. If (as usual) the constant $k$ is positive, then (if $x$ ranges over positive numbers), as $x$ increases, indeed $y$ decreases.

However, there are many other ways that $y$ can decrease as $x$ increases. For example, we could have $$y=\frac{1}{\sqrt{x}},$$ or $$y=e^{-x}.$$ There is no really standard symbolic notation for this, but sometimes arrows are used, as in "as $x\uparrow$, $y\downarrow$." I have also seen slanted arrows used instead, but the standard LaTeX slanted arrows are longer than the arrows I remember seeing.

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1  
$x\nearrow \longrightarrow y \searrow$ ? – Henry Feb 23 '12 at 8:20
    
These look like \nearrow and \searrow, and what I remember seeing was shorter. – André Nicolas Feb 23 '12 at 8:23

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 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(.) is an arbitrary monotonic increasing function".

Finding an intuitive significant for a increasing function is not an easy issue, and perhaps this is the reason why this convention has (in may knowledge) been never established. If not, try yourself to imagine a particular shape associated with increasing.
A first idea I had 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:

  • $Đ(Đ(x))=Đ(x)$
  • $Đ(đ(x))=đ(x)$
  • $Đ(-x)=-Đ(x)$
  • $đ(-x)=-đ(x)$
  • $Đ(x^n)=Đ(Sign(n)x)$
  • etc.

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

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

Of course this is only a proposal that works well for me. If you find it useful, I encourage you to use it, but 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.

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This is an inverse proportionality and physicists (?) are fond of writing them as $x \propto \dfrac{1}{y}$.

Note that this is merely suggestive of the inverse variation and should not be interpreted as a proportionality.

As for $\LaTeX$, here is the code \varpropto for $\varpropto$ or $\propto$ for $\propto$.

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Thanks! Thanks perfect!! I see no difference in the two versions of the symbol though. – Angada Feb 23 '12 at 7:22
    
−1, Anybody who confuses monotony with proportionality should be lapidated with naval fluff. This absolutely non-standard and can be very confusing for readers. – Wrzlprmft Jan 24 at 13:31

I would do $f(t_1) > f(t_0) \forall t_1 > t_0$ or something similar. It is not that hard to write out and is quite clear.

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