7.5: Functions need to be Well-Defined
- Page ID
- 23924
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
- \(\overline{a_{1}}+{ }_{n} \overline{b_{1}}=\overline{a_{2}}+_{n} \overline{b_{2}}\),
- \(\overline{a_{1}}-_{n} \overline{b_{1}}=\overline{a_{2}}-{ }_{n} \overline{b_{2}}\), and
- \(\overline{a_{1}} \times_{n} \overline{b_{1}}=\overline{a_{2}} \times_{n} \overline{b_{2}}\).
Fortunately, these statements are all true. Indeed, they follow easily from Exercise \(5.1.19\):
- 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.
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:
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 \(\left[a_{1}^{b_{1}}\right]_{3} \neq\left[a_{2}^{b_{2}}\right]_{3}\).
Assume \(m, n \in \mathbb{N}^{+}\).
- 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 \([a]_{n}=[b]_{n}, \text { but }[|a|]_{n} \neq[|b|]_{n}\).
- Show that if \(m \mid n\), then there is a well-defined function \[f: \mathbb{Z}_{n} \rightarrow \mathbb{Z}_{m}, \text { given by } f\left([a]_{n}\right)=[a]_{m} .\]
- 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.