Is $\curvearrowright$ a valid symbol for "implies that"?

Question

I learned in high school from my favorite math teacher (who also has a PhD in mathematics) that the $\curvearrowright$ symbol means "implies that" (in German "daraus folgt"; "from that follows").

Now that I am learning higher math elsewhere I have not found this notation anywhere; it always seems to be the $\Rightarrow$ symbol.

The $\curvearrowright$ symbol has really grown on me, and it takes much less time to draw than the commonly used $\Rightarrow$ symbol.

I am just curious if $\curvearrowright$ is also a commonly accepted symbol? Maybe it's an old DDR (communist Germany) thing - as that's where my teacher received his PhD?

Answer 1

The commonly accepted symbols for implications are $\Rightarrow$ and its variation $\Longrightarrow$.

Objectively seen, it does not take much less time to draw $\curvearrowright$ than the commonly used $\Rightarrow$ symbol. The former one uses $15$ and the latter one $10$ letters in MathJax code. Also drawing it on paper does not make a real difference since a short line can be drawn in less than a second.

The only source I could find was in the German Wikipedia article, called "Folgepfeil" (implication arrow):

In TeX werden sie als \Leftarrow und \Rightarrow und \Leftrightarrow (mit dem Großbuchstaben in ausdrücklicher Unterscheidung zum einfachen Pfeil) beziehungsweise \nLeftarrow, \nRightarrow, \nLeftrightarrow (mit vorangestelltem kleinen „n“ für Negation) gesetzt. Auch hier gibt es etliche Varianten:

Then a long table of variations of the implication arrow follows, including $\curvearrowleft$ and $\curvearrowright$.

It is understandable that $\curvearrowright$ has a somewhat personal meaning to you, but I would refrain from using it. The curved arrow is not a commonly used symbol to denote an implication and therefore the usage of this symbol may lead to uncertainty of the reader.

Answer 2

One example: This script* by the german professor Dorothee Haroske uses your symbol all the time.

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* Which I, funnily, looked for just a few days ago because I was searching for a reference of the density of the set of all compactly supported smooth functions in the set of all $L^p(\mathbb R)$ functions, $p\in[1,\infty[$)

Answer 3

I think he may have had something different in mind: imagine that you derive that $x + 1 > 0$ (to take a simple example). From that it follows that $x > -1$.

You sometimes see this written (incorrectly) as

$$x + 1 > 0\ \Rightarrow\ x > -1$$

which is a correct statement, but it doesn't tell you anything about $x$, in the sense that it is still correct when $x = -2$.

If instead of that you write something like

$$x + 1 > 0\ \curvearrowright\ x > -1$$

at the very least you don't write something that has a strictly defined and different meaning, and it is natural to read this as "it follows that" or "hence". I often use $\leadsto$ myself, not sure where I picked that up, but probably during my studies.

In short, $\Rightarrow$ is "implies" and $\curvearrowright$ would be "it follows that".