Modulo Addition has Identity

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $m \in \Z$ be an integer.


Then addition modulo $m$ has an identity:

$\forall \eqclass x m \in \Z_m: \eqclass x m +_m \eqclass 0 m = \eqclass x m = \eqclass 0 m +_m \eqclass x m$


That is:

$\forall a \in \Z: a + 0 \equiv a \equiv 0 + a \pmod m$


Proof

\(\ds \eqclass x m +_m \eqclass 0 m\) \(=\) \(\ds \eqclass {x + 0} m\) Definition of Modulo Addition
\(\ds \) \(=\) \(\ds \eqclass x m\)
\(\ds \) \(=\) \(\ds \eqclass {0 + x} m\)
\(\ds \) \(=\) \(\ds \eqclass 0 m +_m \eqclass x m\) Definition of Modulo Addition


Thus $\eqclass 0 m$ is the identity for addition modulo $m$.

$\blacksquare$


Sources