# 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 $x_i$? "$X$ \ $x_i$"?

• @bordeo Join the two propositions by a conditional: $\forall n\in \mathbb{N}\rightarrow 8^n-5^n\pmod{5}\equiv 0$ Jan 31, 2016 at 21:00
• Thanks man--that looks right! Feb 1, 2016 at 0:19
• @G.Sassatelli Yeah, that looks like a problem. But, literally, that is what my homework says. I am going to double check it... Feb 1, 2016 at 0:21
• @G.Sassatelli Yup, teacher made a typo. Probably should be $8^n - 3^n$, but that is just a guess. Anyhow, thanks for pointing it out. Feb 1, 2016 at 0:23

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 $X \setminus \{x_i\}$.