Definition:Modulo Addition

From ProofWiki
Jump to: navigation, search

Contents

Definition

Let $z \in \R$.

Let $\R_z$ be the set of all residue classes modulo $z$ of $\R$.


We define the addition operation on $\R_z$, defined as follows:

$\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z = \left[\!\left[{a + b}\right]\!\right]_z$

This can be shown to be a well-defined operation.


This operation is called addition modulo $z$.


Comment

Although the operation of addition modulo $z$ is denoted by the symbol $+_z$, if there is no danger of confusion, the symbol $+$ is often used instead.


In fact, the notation for addition of two residue classes modulo $z$ is not usually $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z$.

What is more normally seen is $a + b \left({\bmod\, z}\right)$.


Using this notation, what this result says is:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle a\) \(\equiv\) \(\displaystyle b\) \(\displaystyle \) \(\displaystyle \pmod z\) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle c\) \(\equiv\) \(\displaystyle d\) \(\displaystyle \) \(\displaystyle \pmod z\) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \implies\) \(\displaystyle \) \(\displaystyle a + c\) \(\equiv\) \(\displaystyle b + d\) \(\displaystyle \) \(\displaystyle \pmod z\) \(\displaystyle \)                    

and it can be proved in the same way.

Similarly: $a - c \equiv b - d \left({\bmod\, z}\right)$.


Warning

Compare this with Modulo Multiplication, which is defined only on an integer modulus.


Sources

Retrieved from "http://www.proofwiki.org/w/index.php?title=Definition:Modulo_Addition&oldid=81449"
Personal tools
Namespaces
Variants
Actions
Navigation
ProofWiki.org
ToDo
Toolbox
Google AdSense