7.5: Functions need to be Well-Defined

The discussion of modular arithmetic ignored a very important point: the operations of addition, subtraction, and multiplication need to be well-defined. That is, if $\overline{a_{1}}=\overline{a_{2}}$ and $\overline{b_{1}}=\overline{b_{2}}$, then we need to know that

1. $\overline{a_{1}}+{ }_{n} \overline{b_{1}}=\overline{a_{2}}+_{n} \overline{b_{2}}$,
2. $\overline{a_{1}}-_{n} \overline{b_{1}}=\overline{a_{2}}-{ }_{n} \overline{b_{2}}$, and
3. $\overline{a_{1}} \times_{n} \overline{b_{1}}=\overline{a_{2}} \times_{n} \overline{b_{2}}$.

Fortunately, these statements are all true. Indeed, they follow easily from Exercise $5.1.19$:

1. Since $\overline{a_{1}}=\overline{a_{2}}$ and $\overline{b_{1}}=\overline{b_{2}}$, we have $a_{1} \equiv a_{2}(\bmod n)$ and $b_{1} \equiv b_{2}(\bmod n)$, so Exercise $5.1.19(1)$ tells us that $a_{1} + b_{1} \equiv a_{2} + b_{2} (\bmod n)$. Therefore $\overline{a_{1}+b_{1}}=\overline{a_{2}+b_{2}}$, as desired.

The proofs for $-_{n} \text { and } \times_{n}$ are similar.

Example $7.5.1$.

One might try to define an exponentiation operation by: $$\bar{a} \wedge_{n} \bar{b}=\overline{a^{b}} \quad \text { for } \bar{a}, \bar{b} \in \mathbb{Z}_{n} .$$

Unfortunately, this does not work, because ∧n is not well-defined:

Exercise $7.5.2$.

Find $a_{1}, a_{2}, b_{1}, b_{2} \in \mathbb{Z}$, such that $\left[a_{1}\right]_{3}=\left[a_{2}\right]_{3}$ and $\left[b_{1}\right]_{3}=\left[b_{2}\right]_{3}$, but $\left[a_{1}^{b_{1}}\right]_{3} \neq\left[a_{2}^{b_{2}}\right]_{3}$.

Exercise $7.5.3$.

Assume $m, n \in \mathbb{N}^{+}$.

1. Show that if $n > 2$, then absolute value does not provide a well-defined function from $\mathbb{Z}_{n}$ to $\mathbb{Z}_{n}$. That is, show there exist $a, b \in \mathbb{Z}$, such that $[a]_{n}=[b]_{n}, \text { but }[|a|]_{n} \neq[|b|]_{n}$.
2. Show that if $m \mid n$, then there is a well-defined function $$f: \mathbb{Z}_{n} \rightarrow \mathbb{Z}_{m}, \text { given by } f\left([a]_{n}\right)=[a]_{m} .$$
3. Show that if we try to define a function $g: \mathbb{Z}_{3} \rightarrow \mathbb{Z}_{2}$ by \(g\left([a]_{3}\right)=[a]_{2}), then the result is not well-defined.