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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?

Shaun
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user3187119
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    Is it possible your teacher meant the symbol $\supset$? Apparently both $\implies$ and $\supset$ represent material implications (though I have never seen the latter used in this instance). – Kman3 Jun 13 '21 at 18:53
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    I have never seen that symbol used that way before. – Noah Schweber Jun 13 '21 at 18:53
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    I use $\implies$ (\implies latex command). – Bernard Jun 13 '21 at 18:54
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    I have also seen it as a bilinear product $x \curvearrowright y$ of a $K$-algebra, actually rooted trees algebras. But as an implication only informally (unofficially). – Dietrich Burde Jun 13 '21 at 18:55
  • @Kman3 no, I have had him for many years and I am 100% sure it's ↷ – user3187119 Jun 13 '21 at 18:58
  • does it have to be curved? I use $\rightarrow$ all the time – Asinomás Jun 13 '21 at 19:02
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    It's not standard notation, so I'd discourage using it. Just use $\rightarrow$ which is one of the standard notations for material implication (and quicker to draw too than $\implies$) – Prasun Biswas Jun 13 '21 at 19:03
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    along with the example of @DietrichBurde, another slight word of warning: $\curvearrowright$ is also used occasionally to denote group actions; ie one writes $G\curvearrowright X$ to mean that a group $G$ acts on a set $X$. – Atticus Stonestrom Jun 13 '21 at 19:08
  • @user3187119 If my answer has helped you, I'd appreciate a checkmark :) – vitamin d Jun 17 '21 at 20:59
  • Unforunately not, I've seen the Wikipedia page but haven't seen any additional/historical information around this. – user3187119 Jun 18 '21 at 21:24

3 Answers3

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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.

vitamin d
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  • @user3187119 I'm under the impression that the OP is referring to symbols for implication (meta-symbols), rather than symbols for material implication. Implication is almost universally symbolised as $\implies,$ while as you pointed out, material implication doesn't seem to have a universally-agreed on symbol (I prefer $\rightarrow$). – ryang Aug 01 '21 at 16:08
  • @RyanG Thanks for noticing. Now corrected. – vitamin d Aug 01 '21 at 16:11
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One example: This script* by the german professor Dorothee Haroske uses your symbol all the time.

* 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, $p\in[1,\infty[$)

Maximilian Janisch
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    Very interesting! My teacher was also from Jena, and also studied and got his PhD at the Friedrich Schiller University Jena! Maybe it's a "Jena thing"? – user3187119 Aug 08 '21 at 07:19
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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".

doetoe
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