Modulo Addition has Inverses

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Theorem

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

Then addition modulo $m$ has inverses:


For each element $\eqclass x m \in \Z_m$, there exists the element $\eqclass {-x} m \in \Z_m$ with the property:

$\eqclass x m +_m \eqclass {-x} m = \eqclass 0 m = \eqclass {-x} m +_m \eqclass x m$

where $\Z_m$ is the set of integers modulo $m$.


That is:

$\forall a \in \Z: a + \paren {-a} \equiv 0 \equiv \paren {-a} + a \pmod m$


Proof

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


As $-x$ is a perfectly good integer, $\eqclass {-x} m \in \Z_m$, whatever $x$ may be.

$\blacksquare$


Sources

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