I am trying to define addition and multiplication in the set of all germs of $C^{\infty}$ functions on $\mathbb{R}^n$ at a point $p$.
Before continuing, I would like to pose a quick question: I know that the equivalence class $(f,U)$ is called the germ of $f$ at $p$. So, the set of all germs of $C^{\infty}$ functions on $\mathbb{R}^n$ at $p$ is comprised by several germs (i.e. sets of equivalence classes) that need not be equivalent among themselves, right? For instance, let an equivalence class be $(f,U)$ and another equivalence class be $(g,V)$. Then $(f,U)$ need not be equivalent to $(g,V)$, but both of them, with some other equivalence classes, comprise $C_p^{\infty}$ (i.e. the set of all germs of $C^{\infty}$ functions on $\mathbb{R}^n$ at $p$), right?
Now, let two equivalence classes be $(f,U)$ and $(g,V)$. To define addition and multiplication, one needs to add/multiply the values of the functions $f$ and $g$ at $p$. However, I can not decide the region in which those operations are to be defined? I have read that this is $U \cap V$, but I do not understand why. Can someone elaborate on that?
Thanks!