Definition:Subtraction/Natural Numbers

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Let $\N$ be the set of natural numbers.

Let $m, n \in \N$ such that $m \le n$.

Let $p \in \N$ such that $n = m + p$.

Then we define the operation subtraction as:

$n - m = p$

The natural number $p$ is known as the difference between $m$ and $n$.

Naturally Ordered Semigroup

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

Let $m, n \in S$ such that $m \preceq n$.

By Naturally Ordered Semigroup Axiom $\text {NO} 3$: Existence of Product, there exists a $p \in S$ such that:

$m \circ p = n$

This $p$ is the difference between $m$ and $n$, and denoted $n - m$.

The operation $-$, assigning to $m, n \in S$ with $m \preceq n$ their difference $n - m$ is called subtraction.