# 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 $$\underline{1,3},4,2 \leadsto 1,\underline{3,4},2 \leadsto 1,3,\underline{4,2} \leadsto \underline{1,3},2,\fbox{4} \leadsto 1,\underline{3,2},\fbox{4} \leadsto \underline{1,2},\fbox{3},\fbox{4} \leadsto \fbox{1},\fbox{2},\fbox{3},\fbox{4}$$