What are some useful alternative notations in mathematics?

I've read about Reverse Polish notation where $2+3$ becomes $2\enspace3+$ which gets rid of parentheses. I've seen people using $exp_a(b)$ to denote $a^b$ so it looks more similar to $log_a(b)$, or just writing $f[x]$ to make it clear that $[x]$ is an argument, as opposed to $f(x)$ where $(x)$ can be confused with being a factor. I know Leibniz used to experiment with different notations to see what was most useful, and I think it's good to try out different notation on different problems.

What are some other examples of notation which can help make some problems a little easier to phrase or understand?

• This is a fairly simple one, but often when it is rather clunky to write otherwise, we often write $e^x$ as $\exp(x)$. For example, $\exp(\frac{2\pi i}{n})$ looks a little less clunky than $e^{\frac{2\pi i}{n}}$ at this size. Sep 28 '14 at 21:44

For a function $f:S\to T$ it is very common to write $f(X)=\{f(x)\mid x\in X\}$ for the direct image (for $X \subseteq S$). Similarly, it is very common to write $f^{-1}(Y)=\{s\in S\mid f(s)\in Y\}$ for the inverse image of $f$ (for $Y\subseteq T$). This gives rise to the rather confusing convention that for $f:S\to T$ we get the direct image function $f:\mathcal P (S)\to \mathcal P (T)$ and the inverse image function $f^{-1}:\mathcal P (T)\to \mathcal P (S)$. For some choices of sets the direct image $f$ may conflict with the function $f$ and of course if $f$ is invertible, then we also denote the inverse by $f^{-1}$. Confusing. Some authors use $f_\rightarrow$ for the direct image and $f^{\leftarrow }$ for the inverse image. Notation that is much more suggestive and less confusing.
• In set theory we usually write $f''X$ or $f[X]$ for the direct image. Sep 28 '14 at 22:14