Is there a more concise notation for the following sentence?
"Thus $ f(n) \equiv 0 \pmod{12} $ for all $ n \equiv x \pmod{12} $ where $ x \in \{ 0, 1, 3, 6, 7, 11\} $."
Is this an acceptable notation?
"Thus $ f(n) \equiv 0 \pmod{12} $ for all $ n \equiv 0, 1, 3, 6, 7, 11 \pmod{12} $."
I am curious about it because I have not come across the second style anywhere in any book so far, so I want to know how to correctly write such statements without violating existing conventions.
I think you can do that, but it's not mathematically precise and more a convention. Everybody will understand what you mean and it's shorter than the original statement.
However, you could reduce the amount of numbers further by using the fact that $11 = -1 \mod 12$, so you would only need $n \equiv 0, \pm 1, 3,6,7$.
It's the same as if you write: "Let $A_k=2\cdot k$ for $k=1,...,n$" which is mathematically not perfect, but used everywhere.
Edit: My suggestion is that you use equivalence classes in $\mathbb{Z}/12\mathbb{Z}$ and then write
"Thus $[f(n)] = 0$ for all $[n] \in \{0, \pm 1, 3, 6, 7\}$ where $[.]$ is the equivalence class $\text{mod }12$."
If you write whole sections with those statements you just have to say once, that you're using equivalence classes. Writing "$\text{mod } k$" after each and every congruence is just annoying.