Co-prime Notation

I've never seen it before, but Wikipedia appears to use it. Personally I write $(a,b)$ instead of $\text{gcd}(a,b)$, and so I'd just write $(a,b)=1$. Little confusion will be caused by this notation in a number theoretic context.

Ah, I thought it might be standard notation and wanted the proof to look a bit more professional. Perhaps I should stick to the former if he hasn't taught it this way.

If you're doing something for uni (e.g. exam, coursework), don't introduce notation which isn't in your course unless you explain what it is first. If you were to say "write $a \perp b$ to denote $\text{gcd}(a,b)=1$" then that would be fine, but you shouldn't assume that an examiner will know what you mean if the notation isn't introduced in your lecture notes.

Ah, I thought it might be standard notation and wanted the proof to look a bit more professional. Perhaps I should stick to the former if he hasn't taught it this way.

Professional mathematical writing eschews excessive symbology. It is a common mistake of the amateur to try and abbreviate everything. Nothing looks worse than, for example, statements abbreviated like the following: etc.

Just write '$a$ and $b$ are coprime.'

Why do they teach us to write like that then?

For the purpose of making short hand notes or writing on blackboards. Quantifiers also have their use in defining, say, a predicated set that involves more than one quantifier.

I mean, you were presumably taught at primary school that one can abbreviate 'and' by using the ampersand symbol. I am sure that you were also made aware that in an essay or suchlike, you were still expected to use the full word 'and' rather than '&'. Same rules apply.

I don't think that anyone would suggest that you hand in a piece of written work with, for example, sentences that start with symbols.

Look in any maths book. They mostly use words - not just a collection of symbols. In the statement of lemmata, propositions and theorems in particular, one tends to use symbols and abbreviations only where it saves a lot of space and makes for increased readability.

Our lecturer actually encouraged us to write our proofs using all the shorthand notation - though they're only short proofs. Maybe it's just to make sure we understand what they mean and use them correctly.