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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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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

3 Answers 3

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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$x\nearrow \longrightarrow y \searrow$ ? –  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

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

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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