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 \(x\).

| | | | | created: 2014-06-21 14:18:26 | modified: 2020-06-25 10:07:30 | by: *bookofproofs* | references: [696]

## 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