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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.

Addem
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2 Answers2

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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.

Cornman
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  • Whether or not the quotient ring elements are sets is an artefact of the particular construction of the quotient ring. The goal is to view the elements abstractly - as atomic elements of a new ring whose only essential properties are how the elements are related by the ring operations (so any internal element structure is ignored). For further discussion see the final two paragraphs here. – Bill Dubuque Mar 01 '19 at 19:32
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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.

Bill Dubuque
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