Modular arithmetic: What does $\oplus$ mean in the context of $\mathbb{Z}_n$?

Question

What do $\oplus$ and $\ominus$ mean in the context of modular arithmetic mod $n$, i.e. $\mathbb{Z}_n$? I'm familiar with using it as a bit-wise XOR operator, but in the context of mod $n$, this doesn't sound like it would make any sense.

I am told that this has a modular operator definition and that if $a,b\in \mathbb{Z}_n$ then $(a\ominus b) \oplus (b\ominus a) = 0$. I don't want to know how to show this equality but I include it only give any helpful context about what might be expressed by these symbols.

Answer 1

It is just to differentiate the 'normal' $+$, from the operation with modular numbers.

So: $[a]\oplus [b]:=[a+b]$. Note, that $\oplus$ adds 'sets' and $+$ adds numbers.

Answer 2

Onle likely possibility is that they denote addition and subtraction operations for integers $\!\bmod n,\,$ i.e. $\,a \oplus b = (a\! +\! b)\bmod n,\,$ i.e. perform integer addition $a+b\,$ then return its remainder $\bmod n\,$ (and analogously for subtraction).

It is also possible that they could represent the analogous operations in the quotient ring $\,\Bbb Z/n\Bbb Z.\,$

Without any further context it's difficult to say further.