Nomenclature of random variables $\{X=0, Y=0\}$ same as $\{X=0\}\cap \{Y=0\}$? just a small doubt.
My exercises keep oscillating their nomenclature on this small detail and I always have the other version.
Let $X,Y$ be random variables.
Is $\{X=0, Y=0\}$ the same as $\{X=0\}\cap \{Y=0\}$?
Another example. 
Let $N$ be the number of Users on a webpage. Two files are available for download, one with 200 kb and another with 400 kb size.
$$
\begin{align}
X_n(w) := w_n = \{ & 0:=\text{user downloads no file}, \\
& 1:=\text{user downloads the first file (200 kb)}, \\
& 2 :=\text{user downloads the second file (400 kb)}, \\
& 3:=\text{user downloads both files (600 kb)}\}
\end{align}
$$
I want to express, at least one user downloaded the 200 kb file. Here's how I expressed it
$\{X_1 + X_2 + \cdots + X_n \geq 1\}$. Would this be ok? The book expressed it as $\{X_1=1\}\cup\{X_1=3\}\cup \cdots \cup\{X_n=1\}\cup\{X_n=3\}$. 
Another thing to express: no user downloaded the 200 kb file. I expressed it as
$|\{X_k=1, 1 \leq k \leq N\}|=0$. The book as $\{X_1 \neq 1\}\cap \cdots \cap \{X_n \neq 1\}$. Would my solution be ok?
I'm always in doubt when I'm allowed to use symbols like $+$ and $|\mathrm{modulo}|$ (to get the number of elements). Is this generally always allowed? Many thanks in advance!
Thanks in advance guys!

Your second example is incorrect: if no user downloaded the 200K file, but at least one user downloaded the 400K file, we will still have $X_1 + \dots + X_n \ge 1$.
