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## 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 $$x$$.

## 6.Proposition: Comparing Natural Numbers Using the Concept of Addition

### Bibliography (further reading)

 Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013