Over the years, I have had many questions I left unanswered regarding notation. Forgive the fact that the points in this list are somewhat unrelated, but I thought it best to group them all in one question:
When solving a quadratic equation by factorising, it splits into two equations. For example, $x^2-5x+6=0\implies (x-2)(x-3)=0$. Here, we split the equation into two, to find that $x$ can be either $2$ or $3$. So am I right (notation wise) in concluding $$x=2~~~\veebar~~~x=3$$ where '$\veebar$' denotes an exclusive or. I figured using simply '$\vee$' would make no sense because $x$ couldn't be both at one given time.
When performing row operations (in Gaussian Elimination, for example) what symbol do you place between one matrix and another? A lecturer of mine used to use '$\sim$' meaning 'leads to'. For example: $$\left(\begin{array}{ccc|c}1&2&3&3\\3&5&6&5\\7&8&9&9\end{array}\right)$$ $$\sim \left(\begin{array}{ccc|c}1&2&3&3\\0&-1&-3&-4\\7&8&9&9\end{array}\right) \begin{matrix}~\\-3R_1+R2\\~\end{matrix}$$ and what about the operations themselves? Are they written besides the matrix as done above, or above them? Another lecturer of mine used to write $-3R_1+R_2\longrightarrow R_2$ meaning that $-3R_1+R_2$ is placed in the new $R_2$.
This one is more about typsetting than actual notation. Half of the books I see write $\displaystyle\frac{dy}{dx}$ and $\displaystyle\int\dots\,dx$; the other half write $\displaystyle\frac{\text{d}y}{\text{d}x}$ and $\displaystyle\int\dots\,\text{d}x$. Should the differential operator be typset upright ($\text{d}$) or italicised ($d$)?
I appreciate any insight you could give me on these matters. Forgive my notational pedantry.
