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.