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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 \(x\).

Table of Contents

1. Proposition: Transitivity of the Order Relation of Natural Numbers
2. Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities
3. Proposition: Order Relation for Natural Numbers, Revised
4. Proposition: Every Natural Number Is Greater or Equal Zero
5. Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation
6. 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

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