What are the differences between ( , ), [ , ] and $\langle$ ,$\rangle$ $?$ 
What are the differences between ( , ), [ , ] and $\langle$ ,$\rangle$ $?$

Pardon me for asking stupid questions like this but I'm pretty confused by all these symbols :(
 A: There is absolutely no straight forward difference between them because many times they can be used interchangibly without loss of meaning depending on what nomenaclure one is using. There are however "guidelines" in the losest sense possible.
A: Possible meanings of $(x,y)$ include:


*

*The ordered pair with components $x$ and $y$ (e.g., the 2D point with coordinates $x$ and $y$)

*The open interval of all real numbers strictly between $x$ and $y$

*The scalar product of vectors $x$ and $y$


Possible meanings of $[x,y]$ include:


*

*The closed interval of all real numbers between $x$ and $y$ (inclusive)

*The Lie bracket of vectors $x$ and $y$


Possible meanings of $\langle x,y\rangle$ include:


*

*The ordered pair with components $x$ and $y$

*The scalar product of vectors $x$ and $y$

*The (sub-)group generated by $x$ and $y$


Actually, I suppose that any option listed under one notation is used by at least some authors also by the other notations even if not listed above. I explicitly do not attempt to explain all other possible syntactical possibilities for yet other (widespread conventional) meanings, such as $(x)$, $[x]$, $\langle x\rangle$, $f(x)$, $f[x]$, $A[x]$, $F(x)$, $x\choose y$,  $\left({x\over y}\right)$, $\left[{x\atop y}\right]$, $\left\langle{x\atop y}\right\rangle$, $\left[{x\atop y}\right]$ and whatnot.
