# Integer Addition and Multiplication are Well Defined

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.

• What does it mean for an operation (or for a map in general) to be well-defined? – Alex Wertheim Oct 16 '15 at 4:28
• 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" – tzamboiv Oct 16 '15 at 4:30
• Yes. Do you understand that definition? What are the equivalence classes of $\mathbb{Z}_{n}$? – Alex Wertheim Oct 16 '15 at 4:33
• I'm not exactly sure – tzamboiv Oct 16 '15 at 4:35
• You must understand what the equivalence classes are in this situation. If you don’t, go back and master the matter. – Lubin Oct 16 '15 at 4:39

$a' = a \mod n$ 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.
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 $a \cdot c \equiv b \cdot d \mod n$.