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
