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Proposition: Addition of Integers

Any integer \(x\in\mathbb Z\) is by the corresponding proposition an equivalence class
\[x:=[a,b],\]
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 \(b+d\):

\[\begin{array}{rcl}
x+y&:=&[a+c,b+d],
\end{array}\]

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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