# The meaning of $\rightsquigarrow$ in math?

In order to write a scientific paper, I would like to use the symbol

which look as: $\rightsquigarrow$ or $\leadsto$

I'm wondering about the usual meaning and use of this symbol.

The semantic meaning of $\leadsto$ is literally "leads to". Some possible uses

1. In solving a problem, it denotes "the next step is". For example, sometimes people write $$(x - a)(x-c) = 0 \implies x-a = 0$$ which is technically false. It makes a bit more sense to say that $$(x-a)(x-c) = 0 \leadsto x-a = 0$$ when elsewhere it has already been shown that $x-c \neq 0$. (Strictly speaking using \implies you need to write $$(x-c)\neq 0 \wedge (x-a)(x-c) = 0 \implies x-a = 0$$ to be correct.)
2. When describing an algorithm, the $\leadsto$ symbol is sometimes used to denote the next step, or the next transformation. For example, describing bubble sort I may write $$\underline{1,3},4,2 \leadsto 1,\underline{3,4},2 \leadsto 1,3,\underline{4,2} \leadsto \underline{1,3},2,\fbox{4} \leadsto 1,\underline{3,2},\fbox{4} \leadsto \underline{1,2},\fbox{3},\fbox{4} \leadsto \fbox{1},\fbox{2},\fbox{3},\fbox{4}$$
3. Some logicians have very specific meanings attached to this symbol. Unfortunately my familiarity with respect to such is only "I've seen it in a book."

But the symbol is often also co-opted for other meanings as well. As long as this symbol does not appear frequently in your field, you'll probably be okay if you just clearly define it to mean a certain thing in the beginning of your article and use it consistently.

• @Willlie Thanks so much for your answer. So it seems so scarcely used for a physicist like me, that I may propose an other meaning for the purpose of an article. I would like to use it for a gauge transformation, instead of putting prime on half of the variables, which ends up with cumbersome notations. Would you agree ? Thanks again. Feb 18, 2014 at 13:22
• @Oaoa: I am not quite sure I understand how you want to use it to denote a gauge transformation. Can you give an example? In any case, for the most part as long as a notation is clearly defined, it can be understood by the readers. But in the case where there exists a notational convention for what you want to write already, usually following the convention is better than not following the convention. There are exceptions of course, but usually it is most compelling if the new notation significantly clarifies technical aspects of the computation. Feb 18, 2014 at 13:52
• But "when elsewhere it has already been shown that $x-c \neq 0$", why is it still technically false that $$(x - a)(x-c) = 0 \implies x-a = 0?$$ Feb 29 at 9:51
• @user182601 because that is not an implication that holds for all $x$. (See the parenthetical where I indicated what is the technically correct thing to write.) Mar 1 at 12:10

In topology, especially when used in algebraic geometry, $$\leadsto$$ means "to specialize to". We write $$x_1\leadsto x_0$$, where $$x_0,x_1$$ are two points of a topological space, if $$x_0\in\overline{\{x_1\}}$$, that is, every open neighbhourhood of $$x_0$$ contains $$x_1$$.

In "Set-Valued Analysis" by Aubin and H. Frankowska [1], it appears that $$f : A \leadsto B$$ indicates that $$f$$ is a set-valued map from $$A$$ to $$B$$.

[1] J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Boston: Birkhäuser Boston, 2009. doi: 10.1007/978-0-8176-4848-0.

In graph theory, for vertices u, and v in a directed graph: u⇝v means v is reachable from u.

In other words, there is a path from u to v