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 settheoretic 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 \(x\).
Table of Contents
 Proposition: Transitivity of the Order Relation of Natural Numbers
 Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities
 Proposition: Order Relation for Natural Numbers, Revised
 Proposition: Every Natural Number Is Greater or Equal Zero
 Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation
 Proposition: Comparing Natural Numbers Using the Concept of Addition
Mentioned in:
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
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References
Bibliography
 Kramer Jürg, von Pippich, AnnaMaria: "Von den natürlichen Zahlen zu den Quaternionen", SpringerSpektrum, 2013