Definition:Modulo Subtraction

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

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

where $\eqclass x m$ is the residue class of $x$ modulo $m$.

The operation of subtraction modulo $m$ is defined on $\Z_m$ as:

$\eqclass a m -_m \eqclass b m = \eqclass {a - b} m$

Also denoted as

Although the operation of subtraction modulo $m$ is denoted by the symbol $-_m$, if there is no danger of confusion, the symbol $-$ is often used instead.

The notation for subtraction of two integers modulo $m$ is not usually $\eqclass a m -_m \eqclass b m$.

What is more normally seen is $a - b \pmod m$.