How do I clearly identify the elements of a quotient without the elements of its coset?

Question

Consider the group of integers mod $7$. I think of these as the set $\{0,1,2,3,4,5,6\}$ but I'm aware these represent larger equivalence classes

Now I'm saying things like "these are all less than $7$. But skilled, experienced mathematicians keep saying to me things like "That's not true, because $8$ isn't less than $7$.

Then I say "You know what I mean"

Then they say; "You're not getting it, your statement isn't well-defined".

And I say "I am getting it. But when I say they're all less than $8$, there's a sense in which that is true, and that's the sense in which i mean it."

Then they say "if you're not willing to learn I can't help you" and vote to close and delete.

How in general do I avoid this problem by expressly restricting the domain of discourse to the set I intend?

E.g. if I wanted to say "none of these, cubed, equals any of the others".

The best I can think of is:

Let $X=\{0,1,2,3,4,5,6\}$

Define the map $X\to\Bbb Z/7\Bbb Z$

Then I think can say unambiguously no element of $X$ is a cube of any other. Is there a better or commonly used way?

Answer 1

You can say something like:

To each element $u \in \mathbb{Z}/(7)$ there exists one (and only one) element $n \in \mathbb Z$ with $0\le n < 7$ and $u=[n]$.

You can give that uniquely determined integer $n$ a name, like "the canonical representative of $u$" or "the standard representative of $u$" or something.

And then you just have to be careful to distinguish between $u$ and $n$:

• $n$ is an integer in the set $\{0, 1, \dots, 6\}$, and every element of that set is less than $7$; but
• $u$ is an equivalence class of integers.

The "canonical representative" or "standard representative" establishes a one-to-one correspondence between the elements of $\mathbb Z/(7)$ and the elements of the set $\{0, 1, \dots, 6\}$. But they're not the same thing.