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I am supposed to show that addition and multiplication are well defined in $\mathbb{Z}_n$. I don't have much experience will the term well defined nor experience doing proofs with it. I would appreciate hints related to how to properly prove this property.

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  • $\begingroup$ What does it mean for an operation (or for a map in general) to be well-defined? $\endgroup$ – Alex Wertheim Oct 16 '15 at 4:28
  • $\begingroup$ My textbook defines it as "the operation under consideration is well defined if the result is independent of the representatives chosen in the equivalence classes" $\endgroup$ – tzamboiv Oct 16 '15 at 4:30
  • $\begingroup$ Yes. Do you understand that definition? What are the equivalence classes of $\mathbb{Z}_{n}$? $\endgroup$ – Alex Wertheim Oct 16 '15 at 4:33
  • $\begingroup$ I'm not exactly sure $\endgroup$ – tzamboiv Oct 16 '15 at 4:35
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    $\begingroup$ You must understand what the equivalence classes are in this situation. If you don’t, go back and master the matter. $\endgroup$ – Lubin Oct 16 '15 at 4:39
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$a' = a \mod n$ means $ a' \in \{a_i| a_i= a + jn|$ for any $j \in Z\}$.

So you need to show that this is well-defined. i.e. That for for any $a', a~, b', b~$ where $a' = a~ \mod n$ and $b' = a~ \mod n$ then $a' + b' = a~ + b~ \mod n$ and there is no ambiguity.

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Landau's "Foundations of Analysis" devotes a fair amount of time to this. If you read the "Preface for the Teacher", he talks about how properly defining x+y (and x y) had to be done carefully.

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First of all, in this context it makes sense to think of addition and multiplication as binary operations; i.e. $+: \mathbb{Z}_{n} \times \mathbb{Z}_{n} \rightarrow \mathbb{Z}_{n}$ given by $(x,y) \mapsto x + y \mod n$ and $\cdot: \mathbb{Z}_{n} \times \mathbb{Z}_{n} \rightarrow \mathbb{Z}_{n}$ given by $(x,y) \mapsto x \cdot y \mod n$. In order to show that $+$ and $\cdot$ are well-defined, let $a \equiv b \mod n$ and $c \equiv d \mod n$ and then deduce that $a + c \equiv b + d \mod n$ and $a \cdot c \equiv b \cdot d \mod n$.

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