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Proposition: Addition of Integers

Any integer \(x\in\mathbb Z\) is by the corresponding proposition an equivalence class
where \(a\) and \(b\neq 0\) denote natural numbers representing the equivalence class \(x\). Given two integers \(x:=[a,b]\), \(y:=[c,d]\), \(a,b,c,d\in \mathbb N\), the addition of integers is defined by using the addition of the natural numbers \(a+c\) and \(b+d\):


where \([a+c,b+d]\) is also an integer, called the sum of the integers \(x\) and \(y\). The sum exists and is well-defined, i.e. it does not depend on the specific representatives \([a,b]\) and \([c,d]\) of \(x\) and \(y\).

| | | | | created: 2014-09-21 14:25:51 | modified: 2017-08-17 23:09:45 | by: bookofproofs | references: [696]

1.Proof: (related to "Addition of Integers")

2.Proposition: Addition of Integers Is Associative

3.Proposition: Addition of Integers Is Commutative

4.Proposition: Addition of Integers Is Cancellative

5.Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers)

6.Proposition: Uniqueness of Integer Zero

7.Proposition: Existence of Inverse Integers With Respect to Addition

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Bibliography (further reading)

[696] Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013

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