# Definition: Order Relation for Natural Numbers

Let $$x,y\in\mathbb N$$ be any given natural numbers. The relations "$$>$$" greater, "$$<$$" smaller, and "$$=$$" equal have been defined already from the set-theoretic point of view. These relations can be explained more intuitively by the comparison of natural numbers using the concept of addition explains:

We call the numbers

• $$x$$ equal $$y$$, if and only if $$x=y$$,
• $$x$$ is greater than $$y$$, denoted by $$x > y$$, if and only if there exist a natural number $$u\neq 0$$ such that $$x=y+u$$,
• $$x$$ is smaller than $$y$$, denoted by $$x < y$$, if and only if there exist a natural number $$v\neq 0$$ such that $$y=x+v$$.

Moreover, we say * $$x$$ is greater than or equal $$y$$, denoted by $$x\ge y$$, if $$x$$ is greater than $$y$$, or $$x$$ is equal $$y$$ and * $$x$$ is smaller than or equal $$y$$, denoted by $$x\ge y$$, if $$x$$ is smaller than $$y$$, or $$x$$ is equal $$y$$.

Obviously, by this definition

• $$x$$ is greater than $$y$$, if and only if $$y$$ is smaller than $$x$$,
• $$x$$ is smaller than $$y$$, if and only if $$y$$ is greater than $$x$$,
• $$x$$ is greater than or equal $$y$$, if and only if $$y$$ is smaller than or equal $$x$$,
• $$x$$ is equal than $$y$$, if and only if $$y$$ is equal $$x$$.

Corollaries: 1
Definitions: 2 3
Lemmas: 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15
Propositions: 16 17 18 19 20

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013