Propostion 2.2.12 (Basic properties of order for natural numbers) let \(a,b,c\) be natural numbers. Then
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(a) (Order is reflexive) \(a\geq a\).
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(b) (Order is transitive) If \(a\geq b\) and \(b\geq c\), then \(a\geq c\).
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(c) (Order is anti-symmetric) If \(a\geq b\) and \(b\geq a\), then \(a=b\).
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(d) (Addition preserves order) \(a\geq b\) only and only if \(a+c\geq b+c\).
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(e) \(a<b\) if and only if \(a\pp\leq b\).
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(f) \(a<b\) if and only if \(b=a+d\) for some positive number \(a\).
\(Proof\) See Exercise 2.2.3.