The meaning of $\rightsquigarrow$ in math?

Solution 1:

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.