# Definition:Subtraction/Natural Numbers

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## Definition

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**.