Notation conversion help with respect to combinatorical proof First off, I wouldn't bring this to SO, but my teacher refuses to teach me notation. 
Anyhow...


I'm doing a proof.  The proof says: 
"Show that $8^n - 3^n$ is a multiple of 5 for all non-negative integers $n$. 
How do I say this in notation? 
I've got: $\forall x \in \mathbb{N_0}$ ...and $(8^n - 3^n)\%5=0$...but I don't know the proper way to link this.  
Would I say: Show $\forall n \in \mathbb{N_0}$, $(8^n - 3^n)\%5=0$? (where I join the clauses with a comma)...And is $\%$ even the correct symbol here for the modulo function?
Also, how would I say the set $X$ not including $x_i$?   "$X$ \ $x_i$"?
 A: First, let me say that your teacher might refuse teching you this notation because it is considered bad style in written mathematics. The symbols from formal logic (like $\forall$) should be used nearly exclusively when talking about formulas in formal logic and maybe (carefully!) as a shorthand on the blackboard. Full English sentences are just easier to read. In your personal notes, you can do whatever you want, of course.
If you insist on using symbols anyway, there are many different conventions. A comma is fine, as is a colon. You can also put parentheses around the quantifier or what follows it. For modulo, it is usually used as an equivalence relation and then written $8^n-5^n \equiv 0 \pmod 5$. This leaves us with the following (incomplete) list of possiblities: $$\forall n\in \mathbb N, 8^n-5^n \equiv 0 \pmod 5\\\forall n\in \mathbb N\colon 8^n-5^n \equiv 0 \pmod 5\\(\forall n\in \mathbb N)(8^n-5^n \equiv 0 \pmod 5)$$ which I would all consider correct (but other people might have more pronounced opinions).
For your second question: You should use $X \setminus \{x_i\}$.
